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Local assessment of perturbations

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Local assessment of perturbations
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Hartless, Glen Lawson, 1971-
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viii, 202 leaves : ill. ; 29 cm.

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Thesis:
Thesis (Ph. D.)--University of Florida, 2000.
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Includes bibliographical references (leaves 194-201).
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Printout.
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Vita.
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by Glen Lawson Hartless.

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LOCAL ASSESSMENT OF PERTURBATIONS


By

GLEN LAWSON HARTLESS














A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2000
































Copyright 2000

by

Glen Lawson Hartless


































To Christy













ACKNOWLEDGMENTS

I would first like to thank my chairman, Jim Booth, and cochairman, Ramon

Littell, for all of their help and guidance. Their combined knowledge, insight, and

mentoring has been extraordinary. I would also like to express my gratitude to them

for allowing me to pursue avenues of research that are probably closer to my heart

than theirs. Thanks also go to the other esteemed professors who served on my

supervisory committee: James Algina, Randy Carter, and James Hobert.

I would also like to acknowledge people who have influenced me academically

and professionally over the years: Chuck Nowalk, John Grant, and many others

in the Phillipsburg and Lopatcong school systems, Walter Pirie and the faculty

at Virginia Tech, Ron DiCola at AT&T/Lucent, Larry Leemis at the College of

William & Mary, Danny Perkins, Bo Beaulieu, and many others at the University of

Florida, and also fellow students Jonathan Hartzel, Phil McGoff, and Galin Jones,

among many others. I would also like to thank Allan West for maintaining an

extremely reliable computing environment.

Thanks also go to my parents for their love and support and for instilling in

me the discipline needed to achieve my goals.

Finally, my utmost thanks go to my wife, Christine. I am forever grateful for

her unequivocal love, unending patience, and thoughtful insight.















TABLE OF CONTENTS


page


ACKNOWLEDGMENTS .................


ABSTRACT .... ..... .... ..... .... .. .. .. .. .

CHAPTERS

1 INTRODUCTION TO LOCAL INFLUENCE ............


Prem ise . . . . . . . . . .
Measuring the Effects of Perturbations .
Local Influence Diagnostics .........
Perturbations in Regression .........
Basic Perturbations .............
Brief Literature Review ...........
O utline . . . . . . . . . .


2 COMPUTATIONAL ISSUES IN LOCAL INFLUENCE ANALYSIS 21

2.1 Curvatures and the Direction of Maximum Curvature .... 21
2.2 Local Influence Analysis of a Profile Likelihood Displacement 23
2.3 Building Blocks of Local Influence for Regression ....... 24
2.4 Perturbations to the Response in Regression ........ 27
2.5 Unequal Measurement Precision Applied to Regression . 31
2.6 Benchmarks for Perturbation Analyses . . . . .... ..36
2.7 Local Assessment Under Reparameterizations . . . ... ..40

3 PERTURBATIONS TO THE X MATRIX IN REGRESSION . 49

3.1 Perturbations to Explanatory Variables . . . . ... ..49
3.2 Perturbations to a Single Column . . . . . . .... .. 50
3.3 Perturbations to a Single Row . . . . . . . .... .. 59
3.4 Perturbations to the Entire Design Matrix . . . .... ..65

4 LOCAL ASSESSMENT FOR OPTIMAL ESTIMATING FUNCTIONS 87


Inferential Platform . . . . . . . . . .
Local Assessment of Perturbations . . . . . .
Acceleration, Curvature and Normal Curvature . . .
Algebraic Expressions for Local Assessment . . . .
Computational Formulas for J and P . . . . .


. .. 88
. . 93
. .. 96
. . 98
. .. .103


. . . iv


3
8
3
8
. . . 18
. . . 18
. . . 20








4.6 Local Assessment for Maximum Likelihood Inference . . 110
4.7 First-order Assessment of the Log Generalized Variance . 116
4.8 Local Assessment for Quasi-likelihood Inference ....... .125

5 LOCAL ASSESSMENT OF PREDICTIONS . . . . . .... ..134

5.1 Prediction via First Principles . . . . . . . ... .. 134
5.2 Predictive Likelihood . . . . . . . . . ... .. 136
5.3 Influence Measures for Prediction . . . . . . ... .. 138
5.4 Estimation and Prediction in Linear Mixed Models . . . 140
5.5 Local Influence Methods for Linear Mixed Models . . . 146
5.6 One-way Random Effects Model . . . . . . ... .. 151
5.7 Residual Variance Perturbation Scheme . . . . .... ..154
5.8 Cluster Variance Perturbation Schemes . . . . .... ..159
5.9 Application to Corn Crop Data . . . . . . ... .. 167

6 MULTI-STAGE ESTIMATIONS AND FUTURE RESEARCH . 185

6.1 Local Assessment in Multi-stage Estimations . . . ... ..185
6.2 Synopsis and Additional Future Research . . . . .... ..186

APPENDICES

A MODEL FITTING INFORMATION FOR SCHOOL DATA . . 188

B MODEL FITTING INFORMATION FOR CORN CROP DATA . 191

REFERENCES . . . . . . . . . . . . . . . . ... .. 194

BIOGRAPHICAL SKETCH . . . . . . . . . . . . ... .. 202













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



LOCAL ASSESSMENT OF PERTURBATIONS



By

Glen Lawson Hartless

August 2000

Chairman: James G. Booth
Major Department: Statistics

Statistical models are useful and convenient tools for describing data.

However, models can be adversely effected by mistaken information, poor

assumptions, and small subsets of influential data. Seemingly inconsequential

changes to the observed data, the postulated model, or the inference technique may

greatly affect the scientific conclusions made from model-fitting. This dissertation

considers local assessment of perturbations in parametric and semi-parametric

models. The effects of perturbation are assessed using a Taylor-series approximation

in the vicinity of the fitted model, leading to graphical diagnostics that allow the

analyst to identify sensitive parameter estimates and sources of undue influence.

First, computational formulas are given to assess the influence of perturbations

on linear combinations of the parameter estimates for maximum likelihood. A
diagnostic plot that identifies sensitive fitted values and self-predicting observations

in linear regression is introduced. Second, the eigensystem of the acceleration matrix

for perturbing each element of the design matrix in linear regression is derived.








This eigensystem leads to a diagnostic plot for identifying specific elements that

are influential, and it is shown how the results relate to principal components,

collinearity, and influence diagnostics.

Next, it is shown that the conceptual framework and computational tools

easily extend from maximum likelihood to inference that is performed using

estimating equations. In particular, local assessment of parameter estimates and

standard errors derived from optimal estimating functions is addressed. Algebraic

expressions and computational formulas are given for both first- and second-order

assessment, and applications such as quasi-likelihood are considered.

Finally, the techniques are applied to linear mixed models in order to assess

the influence of individual observations and clusters. While previous applications

considered only estimation, the effects of perturbations on both estimation and

prediction is addressed. It is shown that the effects on the fixed and random effect

inferences can be assessed simultaneously by using an influence measure based on

the unstandardized predictive log-likelihood of the random effects.













CHAPTER 1
INTRODUCTION TO LOCAL INFLUENCE
1.1 Premise

George Box (1982, p. 34) stated "all models are wrong; some models are

useful," and A. F. Desmond (1997b, p. 117) stated "a robust method ought not to

perform poorly in neighborhoods of the nominally true model." Indeed, conclusions

based on a statistical model should be reliable under small perturbations to the

data, the model's assumptions or the particulars of the estimation method. In a

seminal paper, Cook (1986) utilized the basic concept of perturbing the postulated

model or the observed data to develop a variety of diagnostic tools. He showed how

small perturbations can be used to develop easily-computable diagnostics for almost

any model estimated by maximum likelihood (ML) estimation.

1.2 Measuring the Effects of Perturbations

In a statistical analysis, parameter estimates are a crucial part of the

decision-making process. These estimates are a function of the observed data, the

postulated model, and the analyst's choice of statistical inference technique. Any

one of these three aspects of a statistical-modeling situation is subject to uncertainty

and doubt: observed data may be incorrectly recorded or subject to rounding

error, a slightly different model may be just as plausible as the postulated model,

and nuisance aspects of modeling such as choosing hyperparameters are subject to

second-guessing. If small changes in these aspects of a statistical modeling problem

create large changes in parameter estimates, the analysis is unreliable. Therefore,

the change in parameter estimates due to perturbation can serve as a barometer for

the robustness of the scientific conclusions.








Following Cook (1986), a vector of q perturbations is denoted as
w = (WO1, W2,... wq)', and the q-dimensional space of possible perturbations is

denoted as 0. Included in this space is one point, wo, that does not change

the original formulation of the problem, and this is called the null perturbation.

The perturbations are not parameters to be estimated, but rather they are

known real-valued quantities introduced into the problem by the analyst. These

perturbations might represent measurement errors, unequal variances, case weights,

correlated errors, missing predictors or some other change to the data, model or

inference method. Prudent choice of perturbations can lead to useful diagnostics.

For example, if parameter estimates are sensitive to perturbing the contribution

(weight) of a particular observation to estimation, it indicates that the observation

is overly influential on the estimates.

For data y and parameters 0, let L(O; y) denote the log-likelihood from

the original postulated model, and let L,(0; y) denote the log-likelihood resulting

from the inclusion of perturbations. If the perturbation specified is the null

perturbation, then the two likelihood are equivalent. The resulting maximum

likelihood estimators (MLEs) of these two likelihood are denoted as 0 and 06,

respectively. Cook (1986) used the likelihood displacement to summarize parameter

estimate changes under perturbation:

LD(w) = 2[L(0; y)- L(06; y)]. (1.1)

From inspection of (1.1), we see that the likelihood displacement is the difference of

the original likelihood evaluated at two values: the original MLEs and the perturbed

MLEs. Because 0 maximizes the original likelihood, LD(w) is non-negative and

achieves its minimum value of zero when w = w0. As the perturbed MLEs become

increasingly different from the original MLEs, LD(w) becomes larger. Thus, the








size of LD(w) provides a measure of how much the perturbation affects parameter
estimation.
Effects on a subset of parameters, 01 from 0' = (0', 0'), can be obtained by

using the profile likelihood displacement:

LD[I"w) = 2[L(01, 02; y) L(01,, g(06y); y)], (1.2)

where 0 is from = (0 2.), and g(0}1) is the vector that maximizes the profile

log-likelihood, L(01,, 02), with respect to 02. The profile likelihood displacement
allows influence to be "directed" through 01, with the other parameter estimates
only changing in sympathy.
In the next section it is shown how to obtain diagnostic tools from a

local influence analysis of the likelihood displacement and the profile likelihood
displacement.

1.3 Local Influence Diagnostics

Although we could simply choose values for w and compute LD(w), this

is not particularly useful for two reasons. First, LD(w) is often not a closed-form

function of w. Hence, if we wish to find its value, re-fitting the model may be
necessary. Second, even if calculating LD(w) was easy, we must somehow decide

what values to use for w. This choice would be highly data-dependent and a
nuisance for the analyst.
Therefore, we would like a methodology that can examine the information

contained in the function LD(w) and yet does not require a choice of actual values
for w nor involve re-estimating the model parameters. Following Cook (1986),
this can be accomplished using the notion of curvature. Simple plots can then be
used to identify whether estimation is sensitive to any particular elements of the
q-dimensional perturbation.


















1-f -----

Figure 1.1: Schematic of a plane curve with its tangent line. The angle between the
tangent line and the x-axis is .


1.3.1 Curvature of a Plane Curve
Curvature is a measure of how quickly a curve turns or twists at a particular
point (Swokowski 1988). Let (f(t), g(t)) be a smooth plane curve. The arc length
parameter s of such a curve is the distance travelled along the curve from a fixed
point A to another point on the curve (Figure 1.1). Letting tA denote the value of t
that yields point A, s can be computed as

s(t) = (f(u))2 + (gt(u))2)]du, (1.3)

and we may express the curve in terms of s: (f(s-l(s)), g(s-l(s))). Additionally, let
T(s) be the tangent line at a point on the curve, and let q denote the angle between
T(s) and the x-axis.
Curvature is defined as the absolute value of the rate of change in q with
respect to s:

C 0 (1.4)

A plane curve that is turning quickly at (f(t), g(t)) has a larger curvature than a
curve that is nearly straight at that point.








2

1.5


0.5

12
-0.5

-1

-1.5
-2

Figure 1.2: Curve with two circles of curvature



The reciprocal of curvature is called the radius of curvature, and it is the

radius of the best fitting circle at that point on the curve. These circles of curvature

have the same first and second derivatives and tangent line as the curve and lie on

its concave side. Figure 1.2 shows the curve (j sin(t),1 sin(2t)) and two of its circles

of curvature. The one on the left occurs when t = 3 while the other one occurs
2

when t = arccos(O). Their curvatures are 6 and 8/9 respectively. The expression

"turning on a dime" refers to to a sharp turn: one in which the radius of curvature

is small, and consequently the curvature of the path is large.

The formula for curvature is


f'(g"(t) g'(t)f"(t)
C =---------3-.(1.5)
((f'(t))2 + (g(t))2)3

We are particularly interested in assessing curvature at the local minimum of a

function from x to y, i.e. when x = t. For this special case, curvature measures

how quickly the function increases from that minimum value. In this case, (1.5)

simplifies to the acceleration of the curve:

S19X2g) n(
0 = 2g,~ = (1.6)
C x x=xo






















Figure 1.3: Likelihood displacement with directional vectors


where g(xo) is the function's minimum. This result can be applied to LD(w) at w0

when there is only a single perturbation, w. Here, the curvature is given by

O2LD(w) (1.7)
Ow2 Ww=o(1

and it measures how quickly the parameter estimates change when w is moved

away from the null perturbation. A large curvature indicates that even if w is close

to, or local to w0, the likelihood displacement increases substantially. Hence, a

large curvature indicates that even a slight perturbation can be very influential on

parameter estimates.

1.3.2 Using Plane Curves to Describe LD(w)

The curvature of plane curves can also be used to describe LD(w) when

q > 1. Now w can be moved away from wo in different directions, as shown by

the vectors in Figure 1.3 for a two-dimensional perturbation. These vectors can be

written as


LO,(a) = wo+ av,


(1.8)




















Figure 1.4: Intersection of plane and surface creates space curve


where the vector v determines the direction and the value a determines the position
relative to wo. Let P, denote a plane that contains w,(a) and extends directly
upward from 0 to intersect LD(w) (Figure 1.4). The intersection of P" and LD(w)
forms a space curve, (w,(a)', LD(w,(a)))1 Because this space curve is contained
within the plane P,, its curvature can be obtained as if it were a plane curve,

(a, LD(aw,(a)). Thus the curvature of LD(w,(a)) at w0 is simply

Cv 2LD( (a)) (1.9)
Oa2 La=O

Different directions produce curvatures of various sizes. The normalized vector that
produces the largest curvature is denoted Vmax and is of particular interest.

1.3.3 Direction of Maximum Curvature, Vmax
The relative sizes of the elements of vmax indicate which elements of w are
most influential on the parameter estimates. For example, if the ith element of Vmax



1 More formally, these space curves are called normal sections. The term comes
from the fact that each plane, Ps, contains the surface's principal normal vector, the
vector that is orthogonal to the surface's tangent plane at Wo. Curvatures of the
normal sections are referred to as normal curvatures.








is large, then wi is a key player in producing the largest curvature of LD(w) at the

null perturbation, i.e. in producing an immediate change in parameter estimates.

The direction of maximum curvature is the main diagnostic of Cook's local

influence methodology. Typically, Vma is standardized to have length one and is

then plotted against indices 1, 2,..., q. If any elements are particularly large relative

to the others, they stand out in the plot. The analyst then may want to take steps

to alleviate the sensitivity to the perturbation. What, if any, steps to be taken

depend upon the perturbation scheme and the particulars of the data analysis.
For some models and perturbation schemes, the formula for Vmax is known

explicitly, while in other situations it requires some moderate computations. In

analyses such as regression, Vmax is typically a function of familiar quantities.

1.4 Perturbations in Regression

Consider the normal linear regression model:

Y = XO + C, (1.10)

where Y is a n x 1 vector of responses, X is a known n x k model matrix of rank k,

,3 is a k x 1 vector of unknown parameters, and e is a n x 1 vector of independent

identically distributed normal random variables with mean 0 and variance or2 > 0.

Hence, Y N(X ,oa2In), and 0 = (01,...,3k, a2)'.

1.4.1 Scheme 1: Measurement Error in the Response

In order to examine the effects of measurement error in the response,

Schwarzmann (1991) introduced perturbations to the observed data. He considered

fitting a model to the observed data set with slight perturbations: y, + wi, for

i = 1,..., q. Hence, w0 is a vector of zeroes for this perturbation scheme. By letting

q = n, all of the observations can be perturbed simultaneously. This allows the











Table 1.1: Scottish hills race data


Observation Record Ti
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Source: Atkinson (1986)


me (min.)
16.083
48.350
33.650
45.600
62.267
73.217
204.617
36.367
29.750
39.750
192.667
43.050
65.000
44.133
26.933
72.250
98.417
78.650
17.417
32.567
15.950
27.900
47.633
17.933
18.683
26.217
34.433
28.567
50.500
20.950
85.583
32.383
170.250
28.100
159.833


Distance (miles)
2.5
6.0
6.0
7.5
8.0
8.0
16.0
6.0
5.0
6.0
28.0
5.0
9.5
6.0
4.5
10.0
14.0
3.0
4.5
5.5
3.0
3.5
6.0
2.0
3.0
4.0
6.0
5.0
6.5
5.0
10.0
6.0
18.0
4.5
20.0


Climb (ft)
650
2500
900
800
3070
2866
7500
800
800
650
2100
2000
2200
500
1500
3000
2200
350
1000
600
300
1500
2200
900
600
2000
800
950
1750
500
4400
600
5200
850
5000








analyst to determine which observations would have the most effect on parameter

estimates in the event that there is measurement or rounding error.

1.4.1.1 Influence on the full vector of parameter estimates

Atkinson (1986) analyzed a data set of record times for 35 different hill races,

as recorded by the Scottish Hill Runners Association (Table 1.1). Scatter plots

indicate the positive association of the three variables: winning time, race distance

and race climb. (Figure 1.5). The influence of perturbations on the parameter

estimates from a regression of winning time on DISTANCE and CLIMB is of
interest. For convenience, the regressors and response have been centered and scaled,

thereby eliminating the need for an intercept. Thus, the full vector of p parameters
is 0 = (ldist,I Acimb, a2)'.

Figure 1.6 shows vmax plotted against the indices 1 through q = n = 35 for

the Scottish hills data. Element 18, which corresponds to perturbation of the 18th

observation, is the dominant contributor to the direction of maximum curvature.

This observation is from a short race with little CLIMB, and yet it has a large

record time. Similar races have much smaller record times, indicating that race 18

is an outlier. Atkinson (1986) has actually suggested that the winning time for this

race may have been 18.67 minutes rather than 1 hour and 18.67 minutes.

Interestingly, it can be shown that the direction of maximum curvature for

Scheme 1 is proportional to the residuals. Thus, if we standardized the vector
of residuals to length 1 and plotted them against case indices, we would obtain

Figure 1.6. This implies that parameter estimates are most sensitive to a set of

small additive measurement errors when those errors tend to exaggerate (or correct)

the model's error in estimating each data point, with the most exaggeration (or

correction) occurring for the most poorly predicted points.




















.18 1,


043
5 10 15 20 25 30 D~ac


(a) Winning
distance


time vs. race


01)

50 e.


1000 2000 3000 4000 5000 6000 7000 80061imb


(b) Winning time vs. race
climb


Climb
8000
a7
7000
6000
5000
4000
3000
2000 "' *11
*0
1000 J ell
1 5 10 15 20 25 30 Distance
5 10 15 20 25 30


(c) Race climb vs. race dis-
tance


Figure 1.5: Hills data; scatter-plots with highlighted cases


1

0.75
84
0.5

4 0.25
o
0
0

-0.25
-0.5

-0.75


0 5 10 15 20
Case


25 30 35


Figure 1.6: Hills data; response perturbation; Vma. for 0


0


0


a ** ; O0 -






12


I 1
0.75 0.75
0.5 0.5
0.25 a 0.25 *
,.' .. .' .. .. .. '' --- .*,0>* '* "al -- *
i* n. ... *5 'a

-0.25 -0.25
-0.5 -0.5
-0.75 -0.75
1I -1

(a) Vmax for directed influ- (b) vmax for directed influ-
ence on /dist ence on climb

Figure 1.7: Hills data; response perturbation



1.4.1.2 Influence on a subset of parameter estimates

Plotting Vmax for the profile likelihood displacement, LD011 (w), can indicate

which components of w affect a subset of parameters. Examining influence on

91 = a2 results in the same vm,,a as the original likelihood displacement, i.e. it is

proportional to the residuals. The directions of maximum curvatures for 01 = /dist

and 01 = dcimb are shown in Figures 1.7(a) and 1.7(b). The first plot indicates that

small measurement error in race 11's winning time is most influential on A&ist. In the

plot of v.ax for fliimb, elements 7 and 11 stand out. The fact that the two elements

have opposite signs can be used for interpretation. They indicate that maximum

curvature is primarily achieved by perturbing winning times 7 and 11 in opposite

directions (i.e. making one observation smaller and the other larger). Thus, if the

record winning times for races 7 and 11 were slightly closer together or farther apart,

this would affect the slope estimate for CLIMB.

1.4.1.3 Plotting options

Note that the standardized Vma,,, is only unique up to sign change. For

example, Figure 1.8 displays the same standardized vector as Figure 1.6, but with

all of the elements' signs switched. This suggests an alternative plot in which the








1
0.75
0.5
0.25 o o

025 1" 15 ~ 2r O 30 *35
-0.25
O'I
-0.5
-0.75
-1

Figure 1.8: Hills data; alternate rendering of vma.,,



signs of the elements are suppressed: a star plot. Here, each element is displayed

as a point, with the distance from the origin equal to its absolute value. The q

elements are simply displayed around the point (0,0), starting in the first quadrant

and ending in the fourth quadrant. The plot is completed by labelling the points

with indices. Labeling all points can result in cluttered plot (1.9(a)), and so a better

option is to only label the largest elements (1.9(b)). In these renderings, element

18 is again noticeably large, but our attention is also drawn to element 7. This

observation has the characteristics of an influential observation: it has the highest

climb and one of the longest distances (Figure 1.5(c)), as well as the second largest

residual.

Unlike the index plot, the star plot does not give the illusion that the

indices are in a specific order. However, the suppression of element signs can be an

advantage or a disadvantage. The advantage is that when looking at several plots

of directional vectors, suppressing the signs helps the analyst identify patterns by

emphasizing element size. The disadvantage is that the diagnostic interpretation of

the signs is lost.










1- 1-



0.5- 0.5-
7 7

5
2
35 0.5 1 1 0.51
18 13 18
26 31

-0.5- -0.5-



-1. -1-

(a) All elements numbered (b) Largest elements numbered


Figure 1.9: Hills data; IVmaxl as a star plot



1.4.2 Scheme 2: Perturbations to Measurement Precision

Another assumption that can be perturbed is that of equal variance (Cook

1986). This is accomplished by letting V(yi) = for i = 1,... n. Here the null

perturbation is w0 = In.

With this perturbation scheme, we deem that some observations are observed

with more precision than others. The resulting perturbed MLEs weight each case

according to its perturbed variance. If certain cases are influential, then even a

slight down-weighting or up-weighting of them produces large displacements.

This perturbation scheme has an important connection to the widely-used

influence measure Cook's Distance (Cook 1977). Recall that the ih Cook's Distance

is a measure of the change in regression parameter estimates that results from

deleting the ith observation. Clearly a deleted observation provides no information for

finding 3. Similarly, as the ith perturbation is decreased from 1, the ith observation

provides less and less information to the estimation process owing to its increased








variance. Thus, case-deletion can be thought of as a rather drastic perturbation:

one that results in a variance -*0 oo for the ilh observation. Case-deletion is referred
to as a global influence technique because it assesses the effect of perturbations that

are not local to the null perturbation in fl.

1.4.2.1 Influence on the full vector of parameter estimates

Figure 1.10(a) shows the direction of maximum curvature for the Scottish

hills data set. Again, element 18 dominates the direction of maximum curvature.

This implies that the precision of observation 18 is the most influential on the full

set of parameter estimates.

1.4.2.2 Influence on &2

Using the profile likelihood displacement, influence directed upon a2 alone

can be assessed. Observation 18 is again implicated by vmax as being most locally

influential (Figure 1.10(b)).

1.4.2.3 Influence on individual regression parameter estimates

Figure 1.10(d) shows Vma,, when isolating effects on lMist. Although there are

several points that stand out, observation 7 is identified as being most influential

since element 7 is large. This race is also implicated as being locally influential in

the plot of Vmax for /dimb (Figure 1.10(c)). In fact, the DFBETAS diagnostic

(Belsley et al. 1980) also identifies race 7 as being influential on cdiimb. Recall that

the value of DFBETAS$j measures how the jth parameter estimate is affected

by deleting the ith observation. Writing the n DFBETAS for the jth parameter

estimate in vector form, it is shown in Section 2.5 that this vector is proportional to

the following Hadamard (element-wise) product of vectors:


























.. - ...-. . . . -


0.25

10 35
-0.25

-0.5

-0.75

-1


..5 10 6o 15 2 0 25 30 35


(a) Vmax for 6


I !~i~ 2T'T~30 ~
S


(c) Vmax for climbb


(b) Vmax for &2


5 Ct O e0 25 30 35
p


(d) Vmax for fdist


1

0.75

0.5

0.25


S -0.25

-0.5

-0.75
-1


0 9


'.' *,:* S0'''a'5"'3'0 935
Op*


(e) Vmax for / (f) v2 for 4



Figure 1.10: Hills data; variance perturbation


I


5 10 w 15 TOT 75









1
0.75

0.5

0.25


-0.25

-0.5

-0.75
*i

Figure 1.11: Standardized vector of DFBETAS for cdimb





1
1hn rl rlj
1-hil
1
1-h2 r2 r2J (1.11)
1


1- ) f r. rj

Here, hi is the ith diagonal element of the "hat" matrix X(X'X)-1X', ri is the ih

residual, and rij is the ill residual from regressing Xj on the other columns of X.

It can be shown (see Section 2.5) that vma,, for directing the effects of the variance

perturbation on 4j is proportional to


ri rij

r2 r2j (1.12)
r. v L 2


r, rni


The elements of the two vectors differ only by a factor of 1 In fact, standardizing

the vector of 35 DFBETAS for 4c3imb to length 1 and plotting them (Figure 1.11)

produces a figure nearly identical to Figure 1.10(c).








Influence on both regression parameter estimates

Figure 1.10(e) displays the direction of maximum curvature for directed

influence on 6. Plot (f) is the direction of largest curvature under the constraint of

orthogonality to Vma,. The two plots are very similar to the directed influence plots

in (c) and (d).

1.5 Basic Perturbations

As an alternative to finding the direction of maximum curvature, we may

utilize basic perturbations. The i1th basic perturbation is the same as the null

perturbation except for an additive change of 1 in its ith element. For example, the

first basic perturbation is w0 + (1,0,..., 0)', and corresponds to moving a distance

of 1 along the first axis in fl. Typically, the curvatures for each of the q basic

perturbations would be computed and then plotted together.

As an example, consider applying the variance perturbation to a linear

regression, and examining directed influence on J3. In this case, the ith basic

curvature, Cb,, is a local influence analog to the ith Cook's Distance, because it

corresponds to re-weighting a single observation. It is shown in Section 2.7.2.1 that

the ith basic perturbation for this scheme measures influence on the ith fitted value.

Figure 1.12(a) plots the 35 curvatures for the basic perturbations applied to the

Scottish hills data, while Figure 1.12(b) plots the 35 Cook's Distances. Although

the scales differ, they provide similar information. Both plots draw attention to race

7, but they differ somewhat in their emphasis of races 11 and 18.

1.6 Brief Literature Review

Local influence techniques have been applied to a number of statistical

models, most notably univariate and multivariate linear models (Cook 1986; Cook

and Weisberg 1991; Lawrance 1991; Schwarzmann 1991; Tsai and Wu 1992; Wu

and Luo 1993a; Kim 1995; Kim 1996c; Tang and Fung 1996; Fung and Tang 1997;









2.5 .

3 2
1.5
2


0.5
1 3
.p.*. *.. l...i. O,. . ...pg..s... 0...
. 0 15 2O 25 30 35 5 10 15" 02"o 5- 30 3

(a) Basic curvatures (b) Cook's Distances

Figure 1.12: Hills data; Cb's for variance perturbation and Cook's Distances



Kim 1998; Rancel and Sierra 1999). A number of articles have also appeared on

applications in generalized linear models (GLMs) (Cook 1986; Thomas and Cook

1989; Thomas and Cook 1990; Thomas 1990; Lee and Zhao 1997) and survival

analysis (Pettitt and Daud 1989; Weissfeld and Schneider 1990; Weissfeld 1990;

Escobar and Meeker 1992; Barlow 1997; Brown 1997). Other applications include

linear mixed models (Beckman et al. 1987; Lesaffre and Verbeke 1998; Lesaffre et al.

1999), principal components analysis (Shi 1997), factor analysis (Jung et al. 1997),

discriminant analysis (Wang et al. 1996), optimal experimental design (Kim 1991),

structural equations (Cadigan 1995; Wang and Lee 1996), growth curve models

(Pan et al. 1996; Pan et al. 1997), spline smoothing (Thomas 1991), Box-Cox

transformations (Lawrance 1988; Kim 1996b), nonlinear models (St. Laurent and

Cook 1993; Wu and Wan 1994), measurement error models (Zhao and Lee 1995; Lee

and Zhao 1996), and elliptical linear regression models (Galea et al. 1997).

Many authors have presented modifications of, extentions to, and theoretical

justifications for the basic methodology (Vos 1991; Schall and Gonin 1991; Schall

and Dunne 1992; Farebrother 1992; Wu and Luo 1993a; Billor and Loynes 1993;

Kim 1996a; Fung and Kwan 1997; Lu et al. 1997; Cadigan and Farrell 1999; Poon








and Poon 1999). Extentions to Bayesian methodology have also been considered

(McCulloch 1989; Lavine 1992; Pan et al. 1996).

1.7 Outline

In Chapter 2, computational formulas for local influence analysis are discussed

and derived for the two perturbation schemes discussed in this chapter. Chapter 3

contains derivations and examples of local influence diagnostics for linear regression

under perturbations to the X matrix. In Chapter 4, the methodology is extended to

a general class of influence measures and estimation techniques. Computational tools

are also provided. In Chapter 5, inference and local influence analysis for prediction

problems are discussed. Techniques for assessing estimation and prediction in linear

mixed models are developed and applied to data. In Chapter 6, some preliminary

results that can be applied to multi-stage estimations are presented. The chapter

also contains commentary on additional areas of future research. The appendices

contain details for data analyses.













CHAPTER 2
COMPUTATIONAL ISSUES IN LOCAL INFLUENCE ANALYSIS

In this chapter, computational formulas for local influence are reviewed

(Cook 1986; Beckman et al. 1987), To demonstrate the mechanics of the formulas,

detailed derivations of previous results for the regression model (Cook and Weisberg

1991; Schwarzmann 1991) will be presented with minor additions. Next, the issue

of how to obtain objective benchmarks is considered, and some new novel solutions

are outlined. Finally, local influence is discussed under reparamaterizations of the

perturbation space f and the parameter space E. By applying previous results

(Escobar and Meeker 1992; Pan et al. 1997), some new diagnostic tools for influence

on fitted values in linear regression are derived.

2.1 Curvatures and the Direction of Maximum Curvature

Cook (1986) showed that the curvature of LD(w) in direction v can be

expressed as

C, = 2v'A'(-L--)Av, (2.1)

where -L is the observed information matrix of the original model,

-L = L(0; y)


and Apxq has elements

Aij = &2Lw (O; y)
aOiOwj O=6,w=wo

for i = 1,... ,p and j = 1,... q. This result is proved in Chapter 4 as a special case

of Theorem 4.2.








The q x q matrix F = 2A'(-L-)A is the acceleration matrix. It is easily

obtained because A is a function of L,(0; y), and -L-1 is typically computed

as part of ML estimation since it serves as an estimated asymptotic covariance

matrix for the parameter estimates. Hence, computing the curvature in a particular

direction is straightforward. In addition, the diagonal elements of the acceleration

matrix are the q basic curvatures. Finally, it is well known how to obtain the

maximum value of a quadratic form such as (2.1) (Johnson and Wichern 1992,

pp. 66-68). Specifically, Cma, is the largest eigenvalue of F, and Vmax is the
corresponding eigenvector. Hence, finding Cm. and Vmax is numerically feasible.

The acceleration matrix can provide additional information. For example, the

second largest eigenvalue is the largest curvature in a direction that is orthogonal

to Vma. Hence, its corresponding eigenvector indicates a direction of large local

influence that is unrelated to the first one. Additional elements of the spectral

decomposition of F can be used in a similar fashion.
A consequence of this derivation is that it shows how local influence

techniques reduce the dimensionality of examining LD(w) via a small number of

curvatures and directional vectors. It is well known that the number of non-zero

eigenvalues of a symmetric matrix such as F is equal to its rank. Since it is a

product of other matrices, the rank of F is no more than the minimum of their

ranks, which is typically p.

In the regression examples given previously, p = 3 and q = 35, meaning that

the local influence approach can summarize the 35-dimensional surface LD(w) with

three curvatures and directional vectors. Similarly, LD0(1(w) was summarized with

just two, as shown in Figures 1.12(c) and (d).

Despite the advantages of the eigen-analysis of F, one difficulty that arises

is the occurence of eigenvalues with multiplicities greater than one. For example,

if Cmax = A1 has multiplicity greater than 1, then there is no unique Vmax to plot.








In these situations it may be more prudent to look at directed influence on single

parameter estimates.

2.2 Local Influence Analysis of a Profile Likelihood Displacement

The profile likelihood displacement, LD[l](w), was presented in Section
1.2 as a measure of influence directed through 01. Cook (1986) showed that the
acceleration matrix can be expressed as

Po"1 = -2A'(L1 B22)A, (2.2)


where

0 0
B22 =
0 L-12

and L22 comes from the partition

L,= ,12

L21 L22

where

t3 9- 9'
L -00200, o=


This result is proved in Chapter 4 as a special case of Theorem 4.2.
Beckman et al. (1987) considered directing influence on a single parameter

estimate. They noted that in this case, the acceleration matrix is the outer product
of a vector, and hence has only one non-zero eigenvalue. Without loss of generality,
suppose that we wish to direct influence on the first parameter estimate in 0. The
eigensystem is then given by


A= 21|A'TII v oc AT,


(2.3)







1
where T is the vector for which TT' = (L B22). It can be expressed as


T= )1 (2.4)


where c = L1 L12L21L21

2.3 Building Blocks of Local Influence for Regression

In this section, the building blocks of F for a multiple linear regression are

constructed. Subsequent sections apply these results to the response and variance
perturbations used in Chapter 1.

2.3.1 Notation

Unless otherwise noted, the derivations assume the presence of an inter-
cept in the model so that the residuals sum to zero. Centering and scaling

the regressors tends to faciliate interpretation, but is not necessary.

The number of regressors is k = p 1.
H = X(X'X)-1X.

Xi represents the ith column of X, and x\ represents the ith row of X.

r = (In H)y represents the residuals.

r, represents the residuals from fitting a model to the ih predictor using

the other predictors. These can be considered as values of an adjusted

predictor (i.e. adjusted for relationships with other predictors). Note:

Hr, = ri and (I H)r, = 0.

X(i) represents the X matrix with the ith column removed. Similarly,

0(i) is the regression parameter vector with the ith element removed, and

A(i) is A without the ith row.








2.3.2 Acceleration Matrix

The ML estimates of 0 are the same as those of ordinary least squares,

= (X'X)-'X'y ,and the ML estimate of &2 is r. The p x p estimated

asymptotic variance-covariance matrix of the MLEs is

[&2 (X'X)-1 0
01-- 2&
n

Partitioning A' into [A A',2], the acceleration matrix is given by

F = 2A'(_-L)A

= 2&2A (XX')-1A + --44A2A,2.
n


2.3.3 Acceleration Matrix for Directed Influence on a2

The acceleration matrix for directed influence on 6,2 will utilize the matrix

B22, which is given by

0 0
0' 2&
n

This implies that the acceleration matrix for directed influence on 8.2 is

[2] = -2A,( 1 B22)A
46,4 (2.5)
= -A-A2,.
n

Noting that A'2 is in fact a q x 1 vector, there will be a single eigenvalue and

eigenvector given by


n44 11A' V1OCAt.2

These quantities are the maximum curvature and its direction, respectively.








2.3.4 Acceleration Matrix for Directed Influence on 13

When directing influence on /3, we have

B22__2 (XX')-1 0
L _B22 :
Tf 0

The resulting acceleration matrix is


= 2,&2A (XX')-1 Af.


(2.6)


Note that this implies that directed local influence on /3 is equivalent to local
influence for /3 when treating a2 as known and equal to its MLE.

2.3.5 Acceleration Matrix for Directed Influence on a Single Regression Parameter

By applying the general results of Beckman et al. (1987), we can construct T
(2.4) for directed influence on 31 in a regression model. Using results for the inverse
of a partitioned matrix, TT' = -(L B22) can be expressed as


1
*"*2
d -(l)X(1))-I X!)Xl

0


-XX(l) (X(1)X(1))1 0
(X(l)X(l))- X1)XlXlX(1)(X(1)X(1))X 0-1 ,

O-1 0
(2.7)


where d = ,~X 1
where d = XlX1 X'X(1)(X(1)X(1)) X'()Xl.
We denote (X'1)X(1)) X'1)Xl by j'1, as it is the regression parameter

estimates from modeling X, as a linear combination of the other explanatory

variables. This simplifies the notation of (2.7) greatly:


-(L B22) 2
1Iirl12


1 -Vi0

-7i1 7i'1 0
0 0' 0


(2.8)








Using these results, we find

1

T lrll (2.9)
0

and this leads to the following expressions for the maximum curvature and its

direction when directing influence on 13i:


Cmax = 21|A'TI| = 2 |2 A1 A/ iljl vmax oc AT oc A A^

The next two sections apply these results to document more completely the

two perturbation schemes given in Chapter 1.

2.4 Perturbations to the Response in Regression

The results of this section closely follow those of Schwarzmann (1991),

with only minor additions. As in Chapter 1, a set of additive perturbations are

introduced into the observed data to examine the influence of measurement error in

the response. An n x 1 vector of perturbations is introduced such that the perturbed
likelihood is based on Y + w N(X13, a2In). The null perturbation is wo0 = On.

Equivalent results can be obtained by perturbing the model of the mean to be

X3 w. That is, by considering a model assuming Y N(X/3 w, a2In). Hence,

the measurement error perturbation can be interpreted as a type of mean-shift

outlier model; each perturbation is a "stand-in" for an additional parameter that
could correct or exacerbate lack-of-fit.








2.4.1 Building Blocks
The perturbed likelihood is


L,,(O; y) cx -- logu2 -


1(y Xf + Wy Xf3 + w)


n 1
n log 02 W-(y + W)'(y + w)


"i 2
-n log a -
2


1-- (' + 2w' w'w),


where y = y Xf3. The next step is construction of A. Partial derivatives with
respect to w are as follows:


OL,(O;y)0; 1
Ow (2 +2w)
1

1
W-(y XO +W).


Second partial derivatives are then


92 L,,(0; yi)
OUL,(0; y) _

Oaua2


1
- x
U2

1


-X03+w).


Evaluating at the original model yields


902L,(O; y)
OwO/3' -=6,6 =W


a2L,(0; y)
woao2 0=-,=wo


1


1
-r


From these quantities, the n x p matrix A' can be constructed:

A'= X r ]








Finally, the acceleration matrix is

S1 r 2 (XX')-1 0 X'
F-=2- IX[2xt-

2 2H (2.10)rr'

&2 n&4
8.2
2.4.2 Directed Influence on a2

The acceleration matrix for the profile likelihood displacement for &2 is
simply the second term of (2.10):

[2] rr', (2.11)
n&.4

Solving the characteristic equations (F AI)v = 0 directly, the single
eigensolution is

Cmax = Amax = Vmax (c r.

Therefore, as mentioned in Chapter 1, Vmax for directed influence on &2 is
proportional to the residuals,

2.4.3 Directed Influence on/

Without loss of generality, consider directed influence on /13i. Rather than
find the acceleration matrix, we proceed directly to determining the sole eigenvector

proportional to A'T and the corresponding curvature 211A'T112, as per (2.3) and








(2.9):

I [ 1 1
VmaxO cx Xl X(1) i ^1


S [xl- x(l)- ]
&IlrxII
1
iI,111
-< T7--'1
oc ri

and


C 211 1i~ r 112= 2
Cmax =2I _r__ I 2

Therefore, the direction of maximum curvature for directed influence on f3 is

proportional to the jth adjusted predictor.

2.4.3.1 Directed influence on

The acceleration matrix for the profile likelihood displacement for 3 is the

first term of (2.10):

-] H. (2.12)
&~2

This matrix has a single non-zero eigenvalue of with multiplicity k, implying that

there are an infinite number of directional vectors that have maximum curvature.

Although not discussed by Schwarzmann (1991), one set of orthogonal solutions can

be motivated by considering the canonical form of the model:


Y = Za + E,


(2.13)








where

Z=Xp



W= [IW ... k I and

i = the ith eigenvector of X'X.

Because Wp' = Ik, this is simply a reparameterization of the original regression
model. The new regressors, Z1 = X I, ..., Zk = Xpk, are known as the principal

components, and they are mutually orthogonal. In addition, each satisfies the

characteristic equations of F[]:

2- 2I) Z, T (H I)X i (2.14)


=0.

Therefore, the standardized principal component regressors can serve as an

orthogonal set of directional vectors, each of which has maximum curvature. Note

also that the k adjusted regressors, rl, . rk, are also directions of maximum

curvature, but they do not constitute an orthogonal set.

2.4.3.2 Influence on 0

The acceleration matrix F given in (2.10) has two non-zero eigenvalues: 4

and ^, with the second one having multiplicity k. The first eigenvector is oc r,

and the set of k principal components can be used to complete an orthogonal set of

eigenvectors. Comfirmation of these solutions is straightforward.

2.5 Unequal Measurement Precision Applied to Regression

Cook introduced this perturbation in his original (1986) paper on local
influence, and the results in this section come from his work. An n x 1 vector

of perturbations is introduced such that the perturbed likelihood is based upon








y N(Xf3, U2D(1/w)), where D(1/w) denotes a diagonal matrix with ijh element
-. The null perturbation is w0 = In. The n basic perturbations would be the local
versions of the n Cook's distances (Cook 1977; Cook 1979). In addition, the n basic
perturbations applied to LDIOA(w) are local influence analogs of the n DFBETAS
for/3.
A second interpretation of this perturbation scheme comes from the idea

of case-weights. Case-weights are used in an analysis when each observation has
been sampled with unequal probability. Under disproportionate sampling, cases are
weighted by the inverse of their sampling probability in order to obtain unbiased
regression parameter estimates. In the normal linear model, this weighting is
numerically equivalent to perturbing the error variances. Hence, this perturbation
scheme can be used to examine sensitivity to the assumption of random sampling.

2.5.1 Building Blocks
To begin, we develop the perturbed likelihood, which is

2 1
L,,(8; y) oc -j log a2 2a--(y X/3'D~w)y X/3)

nloga2 -_ 1 'D(w).
2 20r2

The next step is construction of A. Partial derivatives with respect to /3 are as
follows:
OL,(0; y) -1 (-2X'D(w)y + 2X'D(w)X)3)
913 2ar
= X'D(w)p,

and the partial derivative with respect to a2 is

OL,,(O; y) n 1
02 2- + D(.rr
,9,T22 (40 v' ()y







Second partial derivatives are then

a2L(O; y) lX'D()

92L(O; y) y)
9a29I 24w

and evaluating at the original model yields:

2L(O;y) = -X'D(r)
9039W' e=,=w &2
92L.(;y) 0; ,.yo 1 1
o2Ow' -,=0, 24 (r r)' = r'sq,

where is the denotes the Hadamard (element-wise) product, implying that rsq is
an n x 1 vector with elements r?. Combining these we can write

A 2- D(r)X jrq


and

P=2&2 1D(r)X (XXy)-1 1X'D^)
454 [[( )
+4 rsqrf (2.15)
_n [ 1 07 8 q
2 [D(r)HD(r) + 1 r..r
P 2n2 r sqr .


2.5.2 Directed Influence on 62
The acceleration matrix for the profile likelihood displacement for &2 is
simply the second term of (2.15):

W 4 1rqrq (2.16)
Solving the characteristic equations directly, the single eigensolution isn
Solving the characteristic equations directly, the single eigensolution is


Cmax 4 ma
Sn&4 vmax oc rsq








This suggests that &2 is most affected by variance perturbations when they are
scaled according to the squared residuals.

2.5.3 Directed influence on ij
Without loss of generality, consider directed influence on /31. Again we will
proceed directly to determining the sole eigenvector proportional to A'T and the
corresponding curvature 211A'T|12 as per (2.3) and (2.9):


Vmiax (X D(r) X[ X (1)
17
S1D(r) [x1 X(1)Yi] (2.17)
1
= --D(r)rl
&Jlr1JJ
ox r rl

and

S=2n 211 r._ 12=I22. (2.18)
cm 1J I I1 rJ11 r=1

This suggests that maximum local influence of the perturbation on /i is
achieved by perturbing cases that have both large residuals and extreme values for
the adjusted covariate. Cook (1986) provided additional details of how this result
compares to the information in a detrended added-variable plot.
Also, we may compare the information in Vma. with that provided by the
DFBETAS diagnostic (Belsley et al. 1980). Recall that the value of DFBETASi,j
is the standardized difference between /3j and j,(i), the estimate when deleting the
ith observation. The formula is given by

SDFBETAS 3j,(i) c- jri,
DFBETALV J r Vr(j) scjl(1 hI )








where r, is the ith residual, cj is the jh column of X(X'X)-', cij is the ith element
of cj, and s2 = 12 is the mean squared-error estimate of a2. The vector c, can be
n
written in terms of the jth adjusted regressor by partitioning (X'X)-1 as per (2.7).
Forj = 1,

[- 1 1 1
Ci X, X(1) ]f --I [P r].


Using this equality, the vector of n DFBETAS for /3j is proportional to

1
1-hil
1
1-h22 rr. (2.19)

1
l-hnn

Since Vma x ocr rj, observations with large leverage stand out more in a plot of the
n DFBETAS for fj than in a plot of Vmax,. This algebraic relationship between
Vma and DFBETAS has not been noted by previous authors.

2.5.4 Directed influence on/
The acceleration matrix for directed influence on 3 is the first term of (2.15):

= j [D(r)HD(r)].


The eigensystem of the matrix does not have a closed-form. However, some idea of
the directed influence of this perturbation on 3 can be ascertained by utilizing basic
perturbations. The ith basic perturbation corresponds to modifying the variance of
only the ith case, and its curvature is the ith diagonal element of F'1 (Cook 1986):

Cb = r2 hii. (2.20)
(7U










1 1
0.75 0.75
0.5 0.5
0.25 0.25
6 S
4 0y $@il 0 a-- -
5. ."- '" 2 3-0 5" "'is 2 5 -0 -' 35
-0.25 -0.25
-0.5 -0.5
-0.75 -0,75
-1 -1

(a) V2 for (b) v3 for


Figure 2.1: Hills data; variance perturbation



This is similar to the ih Cook's Distance (Cook 1977):

1 2h,.
k(1 hi)2s2 It,


2.5.5 Influence on 0

The eigensystem of F is not closed-form, but it can be found numerically.

As an example, consider the perturbation of the error variances for the Scottish

hills data. There are three non-zero eigenvalues for F: 14.82, 4.91, and 0.37. The

eigenvector associated with the largest eigenvalue was plotted in Figure 1.10(a). It

is similar to Vma.,, for directed influence on &2, as given in Figure 1.10(b). The other

two eigenvectors for F are plotted in Figure 2.1.


2.6 Benchmarks for Perturbation Analyses

An analyst using local influence diagnostics will want to know when a

curvature is "too big" and when an element of Vmax is "too big." Calibration of local

influence diagnostics was the topic of much debate in the discussion and rejoinder

to Cook (1986), and there is still no consensus opinion on the subject. Much of the

difficulty lies in the fact that the perturbation is introduced into the problem by the








analyst. Because the technique does not involve hypothesis testing, critical values

are not appropriate. However, there have been attempts to develop rule-of-thumb

benchmarks, as exist for Cook's Distance and other common influence diagnostics.

In this section, we discuss some general difficulties in developing benchmarks

for perturbation analyses, as well as some issues specific to the local influence

approach. Although it is argued that all sensitivity is "relative", some simple

methods for assessing the size of curvatures and the size of the elements of directional

vectors are presented.

2.6.1 The Analyst's Dilemma

As mentioned before, the perturbation is introduced by the analyst, and this

implies that he/she must decide whether the effects of perturbation are important,

much the way a researcher decides what effect size is important. Consider a situation

in which the researcher is actually able to give the size or stochastic behavior of

the perturbation. Here, we may concretely measure the effects of perturbation

by whether or not the conclusion of a hypothesis test changes. Alternatively, the

effects can be measured by whether or not the perturbed estimates are outside an

unperturbed confidence ellipsoid. However, despite having a concrete measure of the

perturbations's effects, assessing the practical relevance of the results still falls to

the analyst.

2.6.2 Calibrating Curvatures in Local Influence

In addition to the general limitations of a perturbation analysis, an analyst

using the local influence approach must also contend with the fact that the behavior

and size of the perturbations are undetermined apart from the basic formulation.

Loynes (1986) and Beckman et al. (1987) noted that curvatures do not have

the invariance properties needed for a catch-all method of benchmarking. This

necessitates a reliance upon relative influence.








2.6.2.1 1-to-1 coordinate changes in fl

Following Loynes (1986) and Beckman et al. (1987), let us consider 1-to-1

coordinate changes in the perturbation space f. That is, a new perturbation scheme

is constructed with elements w! = k(wa) for i = 1,... q, for a smooth function k. The

new scheme's perturbation, w*, will not yield the same curvature as w. Specifically,

C. = (q _2 C,, where w,, denotes the null perturbation value for wi.
Ow=w0 Wo
-2 G2
As an example, consider perturbing the variance of y1 to be 2 rather than 1-.

Here, k(w1) = w?, resulting in new curvatures that are (_01? .i)2 (2w ) =1 = 4

times the original ones. Hence, despite the fact that the two versions of 0 can

produce the same set of perturbed models, the curvature in identical directions will

change. This implies that a universal benchmark for curvatures is not possible.

Indeed, given the arbitrariness of scaling the perturbation, it should come as no

surprise that curvatures cannot be compared across perturbation schemes. However,

the direction of maximum curvature is unchanged under 1-to-1 coordinate changes

in fl.

2.6.2.2 Interpreting curvatures

In this section, a simple way to assess curvatures from the same perturbation
2
scheme is given. First, RC is a Taylor series approximation to LD(wo + av)
2
(Lawrance 1991; Escobar and Meeker 1992). Also, assuming that n is large, an

asymptotic 100(1 a)% likelihood-based confidence region for 0 is given by values

of 0 such that


2(L(; y)- L(O;y)) < Xp,-.

Thus, if twice the curvature in direction v is larger than 2x,1-, then a perturbation

at a distance of one from w0 in direction v moves the parameter estimates to the

edge of the asymptotic confidence region.








To further develop this idea, let us define the size of a perturbation to be
its Euclidean distance from the null perturbation (i.e., 11w Woll). Suppose further

that we have perturbations in two different directions: WA in direction VA and WB
in direction VB. If ICA\ is k times larger than IC\I, then perturbation WB must be

vk times larger than WA to have the same effect on the likelihood displacement.
Equivalently, in order to perturb the estimates to the edge of the confidence ellipsoid,

it would be necessary to travel Vk times farther in direction VB in 0 as in direction
VA. This provides a simple interpretation for the relative size of two curvatures.

2.6.2.3 Internal benchmarks for curvatures

Building on the interpretation given above, curvatures can be benchmarked

based on their relative size to some sort of "average" curvature for a given scheme
and fitted model. For instance, the average of the q basic curvatures for the fitted
model can serve as a benchmark. Alternatively, the average curvature could be
computed by (1) allowing the squared elements of v to have a Dirichlet(al ... aq)
distribution with all ai equal to one, and then (2) finding the expected curvature.
This would be equivalent to finding the average approximate displacement at a

radius of V2i from Wo (Cook 1986). If such a benchmark is difficult to compute, a
Monte Carlo estimate can be calculated. Because these benchmarks are computed
using curvatures from the fitted model, they result in comparisons that are internal
to the fitted model. The conformal normal curvature of Poon and Poon (1999)
would also be an internal way to assess the sizes of curvatures.

2.6.2.4 External benchmarks for curvatures

It is entirely possible that all of the curvatures of a fitted model are large

due to the nature of the data set, and using internal benchmarks will not alert
the researcher to such a phenomenon. To address this, a benchmark that is not
dependent on the observed data is needed. To this end, we might utilize the
expected value of the curvature in direction v, using 0 in place of the unknown






40

parameters 0. The expected curvatures can then be plotted along with the observed

curvatures (Cadigan and Farrell 1999). If the expectation of C.ma is difficult to

obtain, Monte Carlo simulations could be used to estimate it. As an alternative,

Schall and Dunne (1992) developed a means of external benchmarking by developing

a scaled curvature using results on parameter orthogonality (Cox and Reid 1987).

2.6.2.5 Internal versus external benchmarks

Internal and external benchmarks are complimentary tools, since each one

provides different diagnostic information. Comparing curvatures from the same

fitted model highlights relative sensitivity given the data and the design. Comparing

curvatures to their expectations highlights whether the sensitivities are expected

given the experimental or sampling design.

2.6.3 Calibration of Directional Vectors

For a directional vector from the eigensystem of F, the analyst must compare

the individual elements in the vector to determine which aspects of the perturbation

are most important. Typically, it is suggested that the analyst simply look for

elements that stand out as being large relative to the rest. Alternately, a vector

whose elements are all equal in size can serve as a means of comparison. That is to

say, if each of the q elements of the perturbation are contributing equally, then each

is equal to either or --. A particular element whose absolute value is larger

than, say, 2 or 3 times that of equally contributing elements (e.c.e.) would then be

of interest. Hence, the inclusion of horizontal lines at 2- and -- in the plot of

directional vectors may be useful.

2.7 Local Assessment Under Reparameterizations

Pan et al. (1997) show that Vmax is invariant to any 1-to-1 measureable

transformation of the parameter space R. In fact, for 0 = h(k), the entire

eigensystem of Fk, is the same as that of F0. However, directed influence on the







estimates of the redefined parameters can then be examined. Using similar steps to
Pan et al. (1997) and Escobar and Meeker (1992), we can write update formulas
for Ao and Lo and then calculate directed influence on a subset of the redefined
parameter estimates. Using partial differentiation, the building blocks of the
acceleration matrix under the parameterization 0 can be written:

00'
A o= TA (2.21)
A = (-Aoi a

I -o- -) a(2.22)
(900 -' -1 1(00'1
'got ) LO (-50-,

where all partial derivatives are evaluated at the fitted model. From this, the
acceleration matrix for directed influence on 01 can be obtained from the partition

0 = (01, 02)':
F1 -2 = -2A,(L-1 B02)A-
= 00...i 00'
2A' (L B22)a0-AO (2.23)
00'
-2A( (L- -B 2-)AO,

where

0 0
B0^22-
0 L-1
B~22=[022

L22 comes from the partition

Loll L 121
L021 L022

and all partial derivatives are again evaluated at the fitted model.
Although Pan et al. (1997) suggested that (2.23) is invariant to the choice
of 02, they did not provide a proof. To show this result, consider another







reparameterization for which the first pi elements are the same as those of 4:


2 02
This implies that

= I P 0 ((2.24)


and this leads to the following expression for the lower right submatrix of L:


o ao' .. ao a2








where all partial derivatives are evaluated at the original model. Meanwhile, the
acceleration matrix for directed influence on i/1 is
Loo
a -2A O a1' l-1 9
*"[ 122 ,i- ~ ~ i






o V 2 a0(2.26)
t9_..-1 00 0 a0
L022 -





aoq,)
1 90f2 f 2

where all partial derivatives are evaluated at the original model. Meanwhile, the




The invariance of the acceleration matrix for directed influence on to the choices

of 2 is proved by showing that (2.26) is equivalent to (2.23). Using the inverse of
0 L-1
102 (2.26)
ao i 9 0 0 ao,
=-2A(L -9 )A9.
0 "22
The invariance of the acceleration matrix for directed influence on 01> to the choice
Of 0^2 is proved by showing that (2.26) is equivalent to (2.23). Using the inverse of








(2.25),


D0
al'


0 0 D0'
0 L-1 9D
IP22


Substituting (2.27) into


oo 4o 0 1 Dk'.9o'
DO [9 o 0 0 90l 9O'
0 L-1022

L90 O4 0 0 0 02 002 a l


DO0 0 0 O0'
9) lo 9 .2LT-1 L-1 Q0D
L 0 -' 22 0 22 2
DO 0 0 1O'
90' o L-1 90
(2.26) completes the result.
(2.26) completes the result.


(2.27)


2.7.1 Directed Influence on a Linear Combination of Parameter Estimates
As mentioned in Escobar and Meeker (1992), a specific application of these
results is to consider directed influence on a linear combination of parameter
estimates, x'O. There are many ways to reparameterize the model such that x'O is
one of the new parameters. For example, assuming that the first element of x is
non-zero, 40 can be defined as AO, where

A zX(1)
oP_1 IP_1

where x, and x(i) are the first and remaining elements of x, respectively (Escobar
and Meeker 1992). Using equation (2.23), the acceleration matrix for directed
influence on 01 = x'O can be computed, and the resulting diagnostics are invariant
to the choice of 02, .., Op. By using a prudent choice of new parameters, we can
obtain easy-to-compute formulas for the single eigensolution of F [], as per the
following theorem.






44

Theorem 2.1

The maximum curvature and its direction for directed influence on z'O is given by


Cmax = lk2 1IA0LOI2 'max "'-1 X

"-1/2
where ki = L/2x.

Proof of Theorem 2.1

To begin, consider a model with parameterization K' = K'l2 0, where

K ki k2 ... kp]


is a full rank p x p matrix with

Ic -1/
ki = l/2x

kIkj= 0 i/j

k'iki kkl i= 2...p.

By construction,
1
S= X'0 K-1 K'
Ilk,112
000
0 1/2 Ko (90 1 LO1K.


Using the update formulas in (2.21) and (2.22) it follows that





A 1- 10
A4 = -9--




(5- 707) L
L(1=O0 -ILlj, 1(0--


K,11/2ilil/2K = Ilk, 112j_
L 0 0







Next, we show that directed influence on ^ = x'O is not a function of k2,..., kp.
From equation (2.23), we find

x'"-]- 2( Ao i1/2K) (lkl121- lkl112 [ ] ( Klil2Ae)


2 A Loi/2A K 1 ] K'L91'2A
kllk,2 0 00

2 A' iL-1/2 kkl I1/2

2
illill

-lk,112 AoLe xz LO Ao.
(2.28)

The single eigensolution of this rank-one matrix yields the final solution for Cmax
and Vmax. E
The construction of 4 is worth some note. The premultiplication of 0 by
K'Lf/2 orthogonalizes the parameters, in the sense that the off-diagonal elements
of the observed information matrix for 0 are zero. (Note that the orthogonalization
is with respect to the observed information matrix as in Tawn (1987), rather
"1/20
than the expected information as in Cox and Reid (1987).) First, a = Le20
is a parameterization with an observed information matrix equal to the identity
matrix. Then, K is constructed to make orthogonal contrasts of a, with the first
column corresponding to the linear combination of interest. Finally, by scaling the
columns of K to all have the same length as ki, K becomes a scalar multiple of
an orthonormal matrix, and this leads to the algebraic simplifications. Also, if
x'0 = 0j the local influence diagnostics provided by the theorem are the same as
those provided by (2.3), as per Beckman et al. (1987).








2.7.2 Application to Regression
As an example, consider perturbing the variances of a regression model and
examining influence on a linear combination of the regression parameter estimates,
A(x) = x'f3. From (2.28), we find

F iL 1 I (2.29)

= -- D(rX(X'X)-'xx'(X'X)-IX'D(r),
W2h,

where h. = x'(X'X)-lx Further,

Vma c x'(X'X)-1X'D(r) oc Cov(X/3,x'3) -r (2.30)

Cma = 2ix'(X'X)-X'D(r)2 (2.31)
&2h.

where Cov(Xo, x'/3) is the vector of empirical covariances between Xf3 and x'f3.
2.7.2.1 A DFFITS-type measure
Using the variance perturbation and directed influence on z'/3, a local
influence analog of DFFITS can be defined. Recall that DFFITS. is defined as the
externally studentized change in the ith fitted value when deleting the ith observation
(Belsley et al. 1980):

DFFITS. = -
r i]hisi

where p(xi)[ij is the estimate of p(xi) with the ith observation deleted and s22 is the
[i]
mean squared-error estimate of a2 with the ih observation deleted. A large value
for DFFITS indicates that the observation is somewhat "self-predicting". It can be
shown that DFFITS, simplifies to

(1 /h
i,) [i]








A similar measure can be obtained by examining local influence on
p2(xi) = xif/ when using the ith basic variance perturbation. That is, only the
ih observation is reweighted, and the curvature for directed influence on f(xi) is

examined. The curvature for the ith basic perturbation is the ith diagonal element of


2,r~hii (.2
Cb, = BasicFITSi = 2r (2.32)

A large value of BasicFITS indicates that the perturbation of the observation's
precision/weighting greatly influences its fitted value. Thus it is a local influence
analog of DFFITS. Coincidently, BasicFITS5 is equivalent to the ith basic curvature
for the acceleration matrix, as given in (2.20). That implies that for the variance
perturbation, the n basic curvatures of LD(w) are measuring influence on the n
fitted values.
While BasicFITS, is the ith basic curvature when directing influence on /(xi),
we can also define MaxFITS, to be Cma, for directed influence on fA(xi). Using
(2.31),

Cmaxi = MaxFITS. 2 r h "(2.33)
2h Ziihj=

By definition the, ith MaxFITS can be no smaller than the ith BasicFITS. While
BasicFITS provides information about which observation's perturbation influences
its fitted value, MaxFITS provides information about how sensitive each fitted value

is relative to the others.
Example

Consider examining influence on the fitted values for each of the 35 covariate
profiles in the Scottish hills data set. Figure 2.2(a) displays the 35 BasicFITS (i.e.,
each of the basic curvatures for directed influence on the corresponding ft(xi)) .
Race 7 is indicated as being influential on its fitted value, while race 18 is also shown











*


5 10 15 20 25 30 35


(a) BasicFITS


0
S
*0
I

0
0



6

*0




* 00 0


* 00
40


5 10 15 20 25 30 35


(b) MaxFITS


(c) BasicFITS (stars connected by
solid line) and MaxFITS (triangles
connected by dashed line)


Figure 2.2: Hills data; a2D(1/w) perturbation; influence on fitted values




to be quite influential on its fitted value. Plot (b) shows the 35 MaxFITS (i.e. the

Cmax'S for directed influence on each of the 35 /(xi)). Since no point stands out

as being large, the plot indicates that no fitted value is particularly sensitivity to

variance perturbations. Plot (c) shows both sets of diagnostics. The BasicFITS

values are stars joined by a solid line, while the MaxFITS values are triangles joined

by a dashed line. The closeness of the MaxFITS and BasicFITS values for race 7

indicates that this observation is somewhat self-predicting.













CHAPTER 3
PERTURBATIONS TO THE X MATRIX IN REGRESSION

In this chapter, the influence that infinitesimal errors in the X matrix can

have on parameter estimates in a linear regression is addressed. First, previous

results for single column and single row perturbations are reviewed. This is followed

by a generalization of these results to perturbing all elements in the X matrix. It is

shown that the eigensolution of P[ is related to principal components regression,

collinearity, and outliers. Finally, new graphical techniques are presented with the

results applied to the Scottish hills data set and an educational outcomes data set.

3.1 Perturbations to Explanatory Variables

Letting k denote the number of predictors, Cook (1986) proposed that an

nk-sized perturbation can be incorporated into the design matrix in an additive

fashion. These perturbations represent fixed errors in the regressors and should

not be confused with random errors-in-variables techniques (Fuller 1987). The

perturbation scheme can be used to assess the effects of small rounding or truncation

errors in the explanatory variables. Upon perturbation, the X matrix becomes

W1 Wn+l W2n+l ... Wkn-n+1

W2 Wn+2 ... ... Wkn-n+2
X + W3 ... ... ... Wkn-n+3



Wn W2n W3n ... WCkn

The use of a single subscript implies w = vec(W), where the vec operator stacks the

columns of the matrix into a single column. When particular elements of W greatly

affect parameter estimation, it indicates that the corresponding element of X is








influential. This allows the analyst to identify specific combinations of explanatory
variable values that influence estimation. As will be shown in Section 3.4, sensitivity

of the regression parameter estimates to these kinds of perturbations is greater when
there is collinearity amongst the regressors and when the model is calibrated to fit
the observed data too closely.

3.2 Perturbations to a Single Column

The special case of only perturbing a single column of X was discussed
in Cook (1986), Thomas and Cook (1989), and Cook and Weisberg (1991). This
section recreates their results in detail.

3.2.1 Building Blocks

Here and in the following sections, di represents a vector with a 1 in the ith
position and zeros elsewhere. The length of the vector depends on the context of its
use. Note that for di of size k,
1
d(X'X)- d i -rI2 (3.1)

and
1
X(X'X)-ld, = 1, ,r, (3.2)

where r, contains the residuals from regressing X, on the other columns of X.

Without loss of generality, suppose perturbations are added to the n values
of the first predictor (i.e., X1 becomes X1 + w). The perturbed likelihood is

L(; y) oc -n log2 21 (- X'3 A)'y- XP3- '1w)
2 2 (3.3)
=- _logu2 1 2W + 'w).
O2 -o + 0






51

The next step is the construction of A. Partial derivatives with respect to w are as
follows:


9L,(O; y)
(w


=- (y- X 13W).
0-


(3.4)


Second partial derivatives are then


02L,(0O; y)
9w9/31
02L (0; y)
9w 013(l
02L.(O; y)
OwaO2


1
=^(y-x(1)&l)- 2/3i(x,+u,))




- -(y-Xf 1W).
O04


Evaluating at the original model yields


i2Lw(O; y)
0w0fa 0=0,W=Wo
02L,(0; y)
09u90l() O=,w=wo
02L,(0; y)
0wQ^a2 e=eW


1
1 (r 4lX,)

= -x(1)
/41

a4,


and from these quantities we construct the n x p matrix


A' i[r-/l-
=- rd' lX


-]X(1) -Jr]

rI ]


where d, is a k x 1 vector. It follows that the acceleration matrix is given by


(3.5)


dlr' -- 1x
r/


P 21 rd -1 'xx')-i o ]
2 rd _-lx -I] ^01 2&--4
n
=2 + 2- ) rrI + 121H -l(rrl + r)r')







3.2.2 Directed Influence on &2
The acceleration matrix for the profile likelihood displacement for &2 is given
by

F[] = A^,2A (3.6)

n ,rr. (3.7)

Solving the characteristic equations (F AI)v = 0 directly, the single
eigensolution is

Cmax a v max cX r.

Therefore the direction of maximum curvature for directed influence on &2 is always
proportional to the residuals regardless of which column of X is perturbed, but the
corresponding curvature is a dependent upon the effect size, |/3i|.

3.2.3 Directed Influence on/ i
Using the results of Beckman et al. (1987), the sole eigenvector is proportional
to A'T and the corresponding curvature is 2|IA'TI12 as per (2.3):
1 [ ] 5 a1
Vmax 0C r 7 /41X1 -1-X(i) &1[ ]


1 [r 431Xl + 1 1 X] (3.8)

&I1ri|l
Ic r 4r r







and

Cmax 2 r- il'
= 211= 2 1 2'- /1, 112
& 2^jp [rr 2 _ri'+ +4172 (3.9)
=2 1( + Ir2


The direction of maximum curvature is a weighted average of the residuals and the
adjusted regressor. As noted in Cook and Weisberg (1991), 41r1 = y y(), where
Y() is the vector of fitted values when X1 is excluded from the model. Using this
result, we find Vma, oc 2r rl), where r*) is the vector of residuals when modeling
y using all regressors except X1. This indicates that the direction is dominated by
the residuals, a fact which is demonstrated by examples later in this chapter. Cook
and Weisberg (1991) discussed how Vmax relates to de-trended added variable plots.

3.2.4 Directed Influence on Other Regression Parameter Estimates
Thomas and Cook (1989) examined the influence that perturbations in one
column of X can have on the regression parameter estimate corresponding to a
different column. Without loss of generality, suppose we perturb the first column of
X and look at the effects on the last regression parameter estimate /k. Again using
the results of Beckman et al. (1987) we can obtain the maximum curvature and its
direction. Letting di be of size k 1,


vm, oc AT [ (rd, ,X(,)) ] X -f

-i rkI [-r k + ,-H(k)Xk IlXkL (3.10)

ocC ckir Ilrk,







where ikl is the regression parameter estimate corresponding to X1 when predicting
Xk with the other columns of the X matrix. The maximum curvature is given by

Cmax = [i ,2kir'+ ir'k] [kir +OArJ
S icTrk-- k [ 1 + / (3.11)
2 + II- H2 .
1+ Y1 Irk112!
When the partial correlation between X1 and Xk is zero, Cmax achieves its minimum
value of -, yielding Vmax cx rk- In the event that Xk is a linear combination of the
other regressors with non-zero ykl, Cmax = oo and Vmax c r. Between these two
extemes, the direction of maximum curvature is a weighted average of the residuals
and the adjusted kth regressor. These quantities are used to detect outliers and
measure leverage, respectively. If the partial correlation between X1 and Xk is as
large as that between X1 and y, then the two vectors are equally weighted. Note
that this weighting is invariant to scale changes in the columns of the design matrix
as well as the response.
3.2.4.1 Directed influence on /
Without loss of generality, we again consider perturbing the first column.
The rank-k acceleration matrix is

=-- 2 ( 1 rr' + H- (-rr'i + rr') (3.12)


The first eigenvalue of this matrix is unique, and yields a maximum curvature and
corresponding direction of

Cmax = (A2 + Jr ) and vi oc r- lrl.







Hence, the maximum curvature and its direction are identical to that of directed
influence on i. Confirmation of this first eigensolution follows:

-A11) vi (" 4' (i2 + HPE) I) (r _

oc (rr' + _riII2IH I,(rr[ + ,r')) (r ihi)
-(/32r 112 + 11712j1( ~~
-( ,ll+IIl (r ri)
= Itr112r &I 112lri IIl'lII1Hr', + K1r, 112 r
rllr llr 111rr + Illri ll2r, + llrll2,
= -kl,.,lt112Hr, + 'Jll,.11l ri

=0.

The second eigenvalue is /321 and has multiplicity k 1. As mentioned earlier,
this implies that there is no unique set of orthogonal eigenvectors to complete the
solution.

3.2.5 Influence on
The acceleration matrix for examining both f3 and &2 simulatenously has
k + 1 = p non-zero eigenvalues. The form of the eigensolution is unknown. My
numerical studies indicate that there are three unique non-zero eigenvalues, one of
which has multiplicity k 1. The one with higher multiplicity appears to be the
second largest of the three, with a value of'7. ^2

Example
Huynh (1982) and Fung and Kwan (1997) examined data from 20 randomly
sampled schools from the Mid-Atlantic and New England States. The mean verbal
test score (VSCORE) of each school's 6th graders is regressed on staff salaries per
pupil (SALARY), % of white-collar fathers (WCOLR), a socio-economic composite
score (SES), mean teacher's verbal score (TSCORE), and mean mother's educational








Table 3.1: School data; Pearson correlation coefficients

SALARY WCOLR SES TSCORE MOMED VSCORE
SALARY 1.00
WCOLR 0.18 1.00
SES 0.22 0.82 1.00
TSCORE 0.50 0.05 0.18 1.00
MOMED 0.19 0.92 0.81 0.12 1.00
VSCORE 0.19 0.75 0.92 0.33 0.73 1.00


level (MOMED). Appendix A contains model fitting information and diagnostics for

the recession analysis, with all variables centered and scaled.

Diagnostic plots of the residuals, externally studentized residuals, Cook's

Distances, diagonal elements of the hat matrix, and DFBETAS appear in Figure

3.1. Ordinary least squares regression indicates cases 3 and 18 are outliers, although

Huynh (1982) also identified observation 11 as a mild outlier. This data set also

has some redundancy among the predictors SES, WCOLR and MOMED (Table

3.1). Although the X matrix is not ill-conditioned, the association between SES

and MOMED accounts for the unexpected negative slope estimate for MOMED.

Table 3.2 shows the non-zero curvatures (eigenvalues) for perturbing

single columns of the X matrix. Comparing the columns of the table provides

information on which regressor's measurement is most influential. Here we see

that the measurement of SES is typically the most influential, both for regression

parameter estimates and for &2. Comparing the rows provides information on which

parameter estimates are most influenced by these kinds of perturbations. Here we

see that WCOLR and MOMED estimates are the most sensitive. These are the two

regressors that are highly correlated with each other and with SES. It comes as no

surprise that the results for collinear explanatory variables are more susceptable to

measurement errors of this nature.
























9 9
I 9 '
* *S D *1


(a) Residuals


S 10 1iS 20


(d) Hat diagonals


I
0.5 9

*D.*
-0. II 11 20

.1
d.!



(g) DFBETAS for
SES


* t 9s 1 o 10 20
iz 5 "'* U "1'S '
e .



(b) Externally stu-
dentized residuals


9 9 *~ -- *.*. *
~ t~ .91b


(e) DFBETAS for
SALARY


999
9 10 ~ 11
l.1~


(h) DFBETAS
for TSCORE


0.3
0.9


..... .. .. 'D .' o
5 It is 20


(c) Cook's Dis-
tances



Mi
1.5 #
1.
0.51 .9
*, 9 "9 9


0.5

0.1


(f) DFBETAS for
WCOLR


(i) DFBETAS for
MOMED


Figure 3.1: School data; diagnostic plots


* 0 ,


. '. ,.'IV 'fO








Table 3.2: School data; curvatures using X, + w perturbation
Estimates Column being perturbed
in LD(w)
or LD[el](w) SALARY WCOLR SES TSCORE MOMED
0 A1=4.144 A1=20.156 A1=50.008 A1=6.658 A1=19.065
A2=0.440 A2=0.847 A2=19.002 A2=1.413 A2=00.932
A3=0.094 A3=0.071 A3=14.441 A3=0.599 A3=00.091

3 A1=3.357 A1=18.534 A1=26.445 A1=4.432 A1=17.292
A2=0.440 A2=0.847 A2=19.002 A2=1.413 A2=00.932

i 3.357 0.956 19.008 2.126 0.951
/2 0.458 18.534 19.890 1.585 10.869
3 0.443 2.957 26.445 1.501 1.640
/4 1.129 1.859 19.220 4.432 1.295
/5 0.444 11.590 19.324 1.480 17.292

&2 0.881 1.693 38.004 2.825 1.865




Since perturbing the SES variable is most influential, examining directional

vectors for this perturbation is of interest. The element corresponding to case 18 is
the most noticeable in the plot of vmax for the likelihood displacement (Figure 3.2).
This observation has a large residual and is indicated as being influential in the
various diagnostic plots in Figure 3.1. Also note the similarity of Figure 3.2 to the
residuals (Figure 3.1(a)).
Plots (a)-(e) of Figure 3.3 show the directions of maximum curvature when
directing influence on each of the individual regression parameter estimates. In
general the plots indicate different elements of X3 to be influential on different /3.
Finally, Figure 3.3(f) provides the direction of maximum curvature when directing
influence on &2. Recall that this vector is proportional to the residuals.








1
0.75
0.5
0.25
0 0
.5o ft 15 20
o o
-0.25
-0.5 *
-0.75
-1

Figure 3.2: School data; X3 + w perturbation; influence on/3



3.3 Perturbations to a Single Row

The special case of only perturbing a single row of X was discussed in

Thomas and Cook (1989) in generalized linear models. Here, detailed derivations

for regression are presented. The results indicate that single row perturbations have

limited diagnostic value.


3.3.1 Building Blocks

Without loss of generality, suppose perturbations are added to the k regressor

values of the first observation (i.e., x' becomes x' + w'). For d, being of order n,

the perturbed likelihood is

L (O;y) oc log ,2 -1 (y X3 d,'f3'(y X dlw,')
22a
2 1 k k (3.13)
= -log (' 2 y- 2( Owj) + ( jWj)2).
j=1 j=l

Partial derivatives with respect to w are as follows:

'2 (3.(-214+2()j)
9w 212 (3.14)

=1-CE m























0, S 0


5
0 0


1- - -1 .* .41 I 1


'10 n 20
o -0.25

S-0.5

*0.75

*I


(a) vma. for di


1

0.75
S
0.5

o 0.25

15 20
.5 7 2
-0.25

.-


(c) vma for /3


* O000.


15 w


1.g
I


a O **0


5 10 15 020
# I


(b) Vmx for f2


O* O


I
O S


* 5 o 10
O


N .,ioA


(d) Vmax for 04


0 O 5 0 10


O *
o15
-o --
o o


(e) Vmax for /35


(f) Vmax for &2


Figure 3.3: School data; X3 + w perturbation; directed influence plots


o
Oo
B


5
*
I







Since the null perturbation is 0, evaluating at w0 simplifies the expression. We then
proceed to take second partial derivatives:


92L,(0; y) 1
4W49,8i L O a2


0


0
Yi x'f1
0


0


&2L.(0; y) (Y_ -x[03.
aJ9(:r2 W=Oo = ( O4

Evaluating at 0 = 0 and combining the portions for the regression parameter
estimates we obtain


where r, is the first
matrix


a2L .( 0; y) 1 l 1
0wao' 10=,w o Oxi
02 L,,(0; y) 8=,=o r,
L(r2;y) T- 4

residual. From these quantities we can construct the k x p


A'=I


(3.15)


The acceleration matrix is then given by

S. 2 r2 I- 1 [ (OP1 2 I 4 r- '
n hr -
= 2 [(hil + 2r2) 8'r 2 (X'X)-' r1Mx (X'X)-1 ri(X'X)-YXi']
T2 ~~~11rJ21


- Xl0








3.3.2 Directed Influence on 62

The acceleration matrix for the profile likelihood displacement for &2 is given

by

F[J-o ^Q2 [2&14] [2
r n &4J (3.16)

n4r

Solving the characteristic equations (F2] AI)v directly, the single eigensolution is


Ca= ^IIn VmaxOC.

Notice how the curvature depends upon the residual for the case that is being

perturbed, but the directional vector is always proportional to 3 regardless of which
row of X is being perturbed. This suggests that Vmax is not very useful for single

row perturbations. The values of Cmax for each row can be compared to assess which
row is most influential when perturbed. However, the relative effects only depend

upon the residuals, indicating there is little value added by plotting the curvatures.







3.3.3 Directed Influence on f3.
Without loss of generality, we can direct influence on 01:

Vmax oC AT

C [rlIk[ 1


--1 riIk-1 --
I0
-X rl /-'- 1ll


+ (ill Xll)3
hl]


21=|( + -x-2,
Cm.= p-- ,lrl II Y- *[)' + (:ll- X1:0011"2


where il is the predicted value of xll from modeling X1 as a linear combination
of the other regressors. More generally, the maximum local influence on ,Bj by
perturbing row i is

Vmax = ri + (iij xij)O
( -14Y )
Cmx= 2-11-rj II 11(1 ') + (i xij)l2.

If the columns of X have all been standardized, then each of the elements of 7 and
of /3 is less than 1. This suggests that element i of Vmx is likely to be larger than
the other elements. However, the signs of ri and (iij xjj,)/ can work to cancel
each other out. Further interpretation is suppressed in lieu of the more general


Ok

S( rl
Ok-i

rl( 1






64

Cmax
40
35
30
25
20
15
10 oo o*0
S O 0
5 O
i Row
5 10 15 20

Figure 3.4: School data; maximum curvatures for single row perturbations



perturbation presented in Section 3.4. Note that the curvature and directional

vector change if one or more of the columns of X are rescaled.


3.3.4 Influence on 3 and b

The acceleration matrix for directed influence on /3 is


p = -2 hj + rh(X'X)- rlfx(X'X)-1 r(X'X)-1X }. (3.17)


The eigensystem of this matrix is unknown, but my numerical examples indicate

that it has k distinct non-zero eigenvalues. The eigensystem of F for the full set of

parameter estimates is also unknown.


3.3.5 Example

The values of school 18's regressors are most influential when perturbing rows

of X one at a time, as evidenced by a large maximum curvature (Figure 3.4). For

perturbing row 18, vmax is nearly proportional to /3 (Figure 3.5). Note that although

element 3 of Vmax appears quite large, it is actually less than the benchmark of

2 = 894.






65

i
0.75
0.5
0.25

2 3 4 5
-0.25
*
-0.5
-0.75
-1

Figure 3.5: School data; x18 + W', vm for 0



3.4 Perturbations to the Entire Design Matrix

At this point, the general case of perturbing all nk elements of the design

matrix is considered. We use W to denote the perturbations in matrix form:


W1 Wn+l W2n+1 .. W.kn-n+1

W2 Wn+2 ... ... Wk n-n+2

W W w3 ... ... ... Wkn-n+3



Wn W2n W3n ... Wkn

and wj to denote the jth column of W.


3.4.1 Building Blocks

The derivations in this section only require the presence of an intercept in

the model (or centering of the response y). However, for ease of interpretation, we

assume the columns of X have been centered and scaled. The perturbed likelihood

is


L,(O;y) oc log o2 (y- XO W )'(y X3 W3). (3.18)
2 o 20r2





66

Proceeding in a similar fashion to the derivations for perturbing a single column,
partial derivatives with respect to w are as follows:


aL,,(e;y) 1
wJ a2


01(y -
02(Y -


X3) /3WI3
XO) 0wf2


Since the null perturbation is 0, evaluating at wo simplifies the expression:


9L,,(O;y) 1_
9Ow W=o a2


1(y-xf3)
01(y- XJ8)


0k(y XO)


We then proceed to take second partial derivatives and evaluate at

F r 0 ... 0\ t


1
~a2


92 Lw 0; y)0=0
9WO9,3, e0e0,uW=W


0 ...


0:

/31
I2x



&kX


L \
={(Ikr-O3X)
or


92L,(0;y) 1
9w9Oa2 O=Oo,W=Wo -


(3.19)


(3.20)


i3ir
^2r


&3kr


A k(y XJ3) AWO


14(.39r).







From these quantities we can construct the np x p matrix A', as given by Cook
(1986):

A'= [Ikr- X (r)] (3.21)


where denotes the Kronecker product (Searle 1982).
The next step is to construct the acceleration matrix. Here we use the fact
that (A ( B)(C D) = AC 0 BD, assuming comformability of the matrices.
Noting that A = A 1 = 1 A, the acceleration matrix can be written as

2=((I r- 9X (X'X) (Ikr- X')+4 (I'rr'


4+ \n / \

(3.22)

3.4.2 Directed Influence on 62
The acceleration matrix for the profile likelihood displacement for a2 is the
second term of (3.22):

4= Of rr'. (3.23)
n&4

This matrix has a single non-zero eigenvalue, |2, with corresponding
eigenvector oc /3 r. This solution is confirmed below:
[4 4
-I)v oc 4n rr' T l21 ] (03 9 r)

= 40'&rr_11 11].(0r)

4- 0 [ 1 2 1 I M I2 r 11 1 12. 1 r )
=0.







This result implies maximum local influence on 62 is achieved by perturbations in
each column that are proportional to the residuals scaled by the corresponding j3.

3.4.3 Directed Influence on (3
In this section, we derive the full eigensystem of ], where

pp] = 2 (1, 0 r X) ((X'X)-1 0 r' -/3 (X'X)-X').

The following theorem provides the full eigensystem of F for the X + W
perturbation in a regression analysis. Note that eigenvalues in the theorem appeared
in Cook (1986), but the eigenvectors appear here for the first time.
Theorem 3.1
Recall that W. is the jth eigenvector of X'X and that Xoj is the j1th principal
component. Letting 6j denote the jth eigenvalue of X'X, the k non-zero eigenvalues
and eigenvectors of F[ are

S= 2(6n + 2) V a W_-j+ 0 r 3 9 Xpkij+_, (3.25)
) 2L_j+1 -&2 jO kj1 7 ~ kjl


forj = 1,...,k.
Proof of Theorem 3.1
Because the algebraic expressions are so long, the eigensolutions are confirmed in
two steps. Noting that 61 and Okj+1 are the jth eigensolution of (X'X)-1, we
S-j+1






have
] n 11013"12 (9 r)
( -2( 3j- z)^^
2 [(1, (9 r 0 0X) ((X'X)-l (9 r' 2' (X'X)-1X')] (W, (9) r)
T22
2([ + Iim3I2)(W r)
2 Ik (ik r X) (It ^ -j )] 2 J(le + II^H12)(Wj 0 r)
P~ [(Ir-) (IrI2 P) jjl12lI2)~0) (.6
2 [117112 r - X X^)1 -pI 2 Il2+ i2)( 9 r)
&2 ij T 6j
3326

=2 liiir ^ ^ li ( x )].
2 l02 W 9 r 1 f ( 0 ( 9 X j
The following matrix multiplication yields the same expression as (3.26):



-= [(k 9 r X) ((X'X) -' (X'X) -') (3 X Xv?)
&2 (llrli
2- 2(1 + 1012)(0 ) Xpj)

2 [(Ik9 r 0 X) (0- _|13||2Wj)] 2(1!E + 10112)0 Xyj)
= 2 [m11{-i_ r + ( Xa)] 2 ( !l + IIii2)( Xp)
&2 2 3
= 2-1,1 r) 1112O (9 X\)]
(3.27)
Using (3.26) and (3.27), confirmation of the jth eigensolution is immediate:

AI 2= 2n-- + 111)1 (Wk-j+l 9 r)
-26-j+l &2
-_[, 2( n +x 11112) 0( X k- (3.28)


=0.







Finally, for j : j*, the orthogonality of the solutions follows:

(W, (9 r i3 (D X^p)'(j., (9 r (g Xvi.)
= j, 9 r'r ', ( 9 ,jX'r w ,r'X j, + '3 x'XX j,
= 0- 0- 0+f0 +

=0. 0



3.4.3.1 Interpreting Theorem 3.1
The closed-form eigesolution in Theorem 3.1 gives insight into how slight
changes to the observed explanatory variables affect inference. To interpret the
theorem, consider maximum curvature, as per (3.25):

Cmax = 2( + ). (3.29)

The eigenvalues of X'X are traditionally used to assess the amount of collinearity
in the regressors. If 8k is small, there is redundancy in the explanatory variables,
and this causes the inversion of X'X to be unstable. If this inversion is unstable,
so too are the regression parameter estimates, 3 = (X'X)- X'y. The first term in
Cma,,x is large when 5k is small, giving further evidence that the estimates are more
sensitive when collinearity is present. The second term of Cma, is also interesting.
It can be considered a naive measure of goodness of fit; when the relationships
between the explanatory variables and y are strong, |11&32 is large and &2 is small.
Hence, "better fitting" models are more sensitive to perturbation. However, 11'3112/&2
typically grows whenever additional variables are added to the model, indicating
more complicated models are also more sensitive to pertubation. In fact, HII13I2/&2
bears some similarity to R2:

R SSeg IIX0ii2 IIX 112
SSy y n(&2+IIX + 12)








The quantity R2 is the proportion of variance explained by the model, and serves
as a measure of goodness-of-fit. However, R2 can be made artificially large by
adding additional explanatory variables, even when they contribute very little to the
explanatory value of the model. The quantity 11 1H2/&2 behaves similarly: additional
variables decrease the denominator and usually increase the numerator. Of course,
the scale of the regressors is a major factor in the size of |I1|12, indicating that
scaling the regressors would be prudent for interpretation.
So, we may interpret Cmax in the following way. The vector of parameter
estimates is more sensitive when (a) there are collinear variables, (b) there are a large
number of explanatory variables, and (c) the model closely fits the data. We are not
surprised to see that the first two phenomenon are sources of sensitivity. However,
the suggestion that better-fitting models are more sensitive to measurement error
may at first be disconcerting. When viewed in light of Type I and Type II errors,
this is not the case. Recall that a Type I error occurs when a test is wrongly
declared significant, while a Type II error occurs when a test is wrongly declared
insignificant. Generally, a Type I error is considered the worse of the two. So, if
there are strong relationships between X and y, the formula for Cma,, implies this
might be missed because of measurement error in the regressors. However, if there
is a weak relationship, perturbations to X would not make the relationship appear
strong. In this sense, the regression model is better protected against Type I errors
(concluding there are strong relationships when there are none) than Type II errors
(concluding there are weak relationships when in fact there are strong ones) when
there is measurement error in the regressors.
As a final comment on Cma, the term 111112/&2 may be better described as
an indicator of a closely fitting model rather than a well fitting model. This suggests
that if we construct complicated models to fit the nuances of a data set, the resulting








parameter estimates are more sensitive to rounding and measurement errors in the
explanatory variables.
Now let us consider Vma as per Theorem 3.1:

Vmax OC Wk 9 r 3 X( k. (3.30)

This vector and the other eigenvectors of Fl provide detailed information about
which elements of X most affect the stability of the MLEs. The expression for
Vma, shows how outliers and leverage in the final principal component regressor
determine precisely what kinds of mismeasurements have large local influence on f3.
The directional vector can be used to scale W to achieve maximum local influence:

Wmax 0C [9klr lXfk Vk2r 3XXok ... Vkkr &kXp ]. (3.31)

The elements in the first column are a weighted average of the residuals (r) and the
regressor values of the last principal component regressor (Xpk). The respective
weights are the first original regressor's contribution to the final principal component

(kl) and the negative of the first regression parameter estimate (-i).
Alternatively, we can think of the elements in the first row as being a
weighted average of the explanatory variables' contributions to the final principal
component (W) and the negative of the parameter estimates (0'). The respective
weights are the first residual (ri) and the negative of the first observation's value for
the final principal component (-xlk).
3.4.3.2 A special case
In order to examine in more detail how the perturbations given in (3.31)
affect /, we consider a simple situation for which there are closed-forms for the
perturbed parameter estimates. Let us assume that the response and a pair of
regressors have been centered and scaled to have length one, and that the regressors







are orthogonal. The implications of these simplifications are as follows:

|iy||= I|X1 = |Ix211 1 r'r= 1-/1-A2
1 = xy 2 =x'2Y
61= 6 =1 = 12.

The first two eigenvalues of F are the same:

Cma.A, --l- A2 = 2(n- + 2n 1 )1
&2i Pi2 22

and we may choose either of the following vectors for Vma:

r- iXl -lX2
-42-X1 r' 42-X2

Without loss of generality we choose the first one. Letting X" denote the perturbed
design matrix, we have
Xo = X + aW
X=X+aWr-X -X]
X+aIr 4X, (3.32)

= ar + (1-aIai)Xi X2-a 2X1

Xl,.w X2,w ]

for some real number a. Now, consider the effects of perturbation on 41. Assuming
a no-intercept model is still fit after perturbation, the perturbed estimate is given by
r-,i Y (3.33)
ll 1 1 11







where ri, is the residuals from fitting X,,, as a function of X2,". The building
blocks needed are as follows:
1 X
H2, = X2,(X, X2,)X -1 a/2X)(X'2 a2X'1)
1 a22(X
r,,, = (I H )X = (1 a/ acf2)Xi + cX2 + ar

ri,,y = (1 a~l ac42)41 + 42 + a(l 412 /22)
I|ri,,|2 = (1 a ac42)2 + c2 a2(1 2 2)

where c = a.2(1-a$. It follows that
2+a2p
01, a ( I ac2)41 + c42 + a(l -1 2 -_ 2)
(1 a: ac/32)2 + c2 + a2(1 -12 2)

For the sake of comparison, we can also consider perturbing only the first
column. Perturbation in the direction of Vmax when directing effects on /31 results in
the following perturbed X matrix:

X, = X + a*W*

= X+a* Ir- IX 0 (3.34)

= [ a*r + (1 a*I )Xi X2]

as per (3.8). Here, the perturbed estimate is given by

&W a ) + a* (1 -1- 2)
(1a )2 + a'2(1 _/ 2_ 22)"

In order for the two perturbations to be comparable, they should be equal in size.
This is accomplished by stacking the columns of the perturbation matrices and
making the resulting vectors equal length. In other words, the Froebenius norms
of aW and a*W* should be equal. Letting a* = a provides the necessary
adjustment.








Now we can plot &,, for some selected values of /p and I2, considering values
of a ranging from, say, -V2 to vf2-. This eliminates perturbation matrices with
Froebenius norms that are larger than that of the unperturbed X matrix. Figure 3.6
shows plots of &, for four sets of parameter estimates. The solid lines represents the
parameter estimate in the direction Vmax for the two-column perturbation, while the
dashed line represents parameter estimates in the direction Vsmax for the one-column
perturbation. All of the influence curves in plots (a) and (b) have a value of .2 under
the null perturbation, i.e. when a = 0. Likewise, the curves in plots (c) and (d) are
all equal to .5 at the null perturbation. In all plots we see that the single column
perturbation does not have the same effect as the dual column perturbation. The
discrepancy between their effects is larger when /32 is larger (plots (b) and (d)).

3.4.4 Influence in Principal Components Regression
Consider the canonical form of the model, as given in Chapter 1:

Y = Za + 6, (3.35)

where Z = XV and a = p'/3. The following theorem proves that the direction of
maximum curvature is the direction of maximumum directed influence on &k.
Theorem 3.2
Let & denote the MLEs of a for model (3.35). The pairs Aj, vj given in Theorem
3.1 are the maximum curvature and direction of maximum curvature for directed
influence on &k-j+1, j = 1,..., k.































-1 0,5-'


0.5 1


(a) Influence curves for /3 = .2,
/2 =.4


Bi


-1 .-e:~


0.5 1


(b) Influence curves for /l = .2,
/32 = .7


B1


0.5 1


(c) Influence curves for/3i = .5, (d) Influence curves for /3 = .5,
/2 = .4 2 = .7


Figure 3.6: Effects of single and double
double column; dashed line- column 1 onl


column perturbations on Q\. Solid line-


..............








Proof of Theorem 3.2 We begin by obtaining update formulas for A and L1 as
per Section 2.7:

A" = A0/
A/ A^
=App (3.36)
= T,(Ifcr-f3X)p
= ( k 0 r - ,x )3
1

1= (W L r X)



&2 (3.37)
= D(6

= &2D(1/6).








Next, we find maximum directed influence on dk-j+l using the results of Beckman

et al. (1987):

Vmax 0C AT _,Wk+l

0


0
1 r&i -
oc &2 W^ r XV] N jk-j+1

0


0

oc Wk-j+l r (- 9 X k-j+l



Cmax 2| 2 (IAaT _|j+l 2
)(-1 9 r' 0 k-j+1X')(Wk-j+1 0 /3 0 X kj+l)



2 (n&2 1111 blI2k 3+l)
&26k_j+l
2( 6k +1
_k-j+1 ar2

Since these expressions are the jth eigensolution of FP the proof is complete. 0
This theorem gives added insight into the direction of maximum curvature.

In short, Vmax is the direction that changes &k the most. Similarly, the jth largest
eigenvalue and associated eigenvector specifically correspond to effects on &k-j+l.
Also, the sizes of the curvatures show that the estimated coefficients for the most
important principal component regressors are also the least sensitive.








Example 1

Consider the Scottish hills data set described in Chapter 1. The purpose of
this example is twofold- to demonstrate how the X-matrix results can be displayed

graphically, and to demonstrate how collinearity affects the curvature values when

examining influence on f3.
Figure 3.7 gives the direction of maximum curvature for directed influence on

a2. The first 35 elements correspond to perturbations of the DISTANCE regressor

values, while elements 36-70 correspond to perturbations of the CLIMB regressor
values. The value for race 18's DISTANCE is the largest contributor to this
influential perturbation. Elements 7 (DISTANCE for race 7) and 53 (CLIMB for

race 18) are also large

The information in Figure 3.7 can be difficult to discern because the

perturbation vector enters into the likelihood in the form of a matrix, W. Another

way to display Vma is to first take its absolute value and then to consider the

elements corresponding to each column of W. Each portion can then be plotted

against indices 1,...,n and shown together (Figure 3.8(a)). This graphical display

provides a "profile plot" of each column of W's contribution to the direction of

maximum curvature. Here we see again that elements corresponding to observations

18 and 7 as being the most influential. There is also some evidence that measurement

error in the DISTANCE values (solid line) are more influential than error in the

CLIMB values (dashed line).

Figures 3.8(b) and 3.8(b) provide profile plots of the absolute value of the
two directional vectors corresponding to v, and v2 of F Il. The first plot draws

attention to observations 7, 11, and 18, while no observations appear noteworthy in

the second plot.

At this point we turn our attention to the curvature values that accompany
these directional vectors (Table 3.3). The first column shows the curvatures for local








Table 3.3: Hills data; curvatures using X + W perturbation. Model I contains
DISTANCE and CLIMB. Model II contains DISTANCE and CLIMB-RATE
Estimates in LD(w)
or LD[O1](w) Model I Model II

2 Al = 30.47 A = 28.52

3 A1=21.15 A1=16.41
A2=16.49 A2=16.25


influence when fitting the regression model for winning time as a linear combination

of DISTANCE and CLIMB. However, as was evident in Figure 1.5(c), the two

regressors are linearly related. An alternative model can be fit by replacing climb in

feet with climb in feet per distance travelled, CLIMB-RATE This regressor does

not have the strong positive association with distance that the original coding of

CLIMB has. This should reduce the collinearity in the model, thereby reducing

sensitivity to perturbations in the X matrix.

Using standardized regressors, the new model is fit. Column 2 of Table 3.3

shows the curvature values for this new model. Cma.x for directed influence on f

has been substantially reduced, indicating that the MLEs for the second model are

indeed less sensitive to perturbations in the X matrix.

Figures 3.9(a), (b) and (c) provide the profile plots of the absolute value

of the directional vectors that accompany the curvatures in Table 3.3, Column II.

All three plots tend to indicate DISTANCE values as being more influential than

CLIMB-RATE values, and all three plots show substantial change from those in

Figure 3.8. The plot for &2 now indicates the DISTANCE value for observations 7

and 18 as influential, rather than just 18. No elements stand out in Vmax, while case

11's DISTANCE value is a very substantial contributor to v2.








1
0.75
0.5
0.25

S10. 20 .*.30 *'4-0 *5'- **69 *. W
-0.25
-0.5
-0.75
-1

Figure 3.7: Hills data X + W pert.; vma for &2



Example 2

We now consider applying the X + W perturbation to the school data and

examining directed influence on (3. The purpose of this example is to demonstrate

alternative methods for displaying Vma,, when the number of regressors and/or

observations is large.

Figure 3.10 is the plot of Vma. for directed influence on /3. Elements 1-20

correspond to SALARY, elements 21-40 correspond to WCOLR, elements 41-60

correspond to SES, elements 61-80 correspond to TSCORE, and elements 81-100

correspond to MOMED. The two regressors that have little association with other

explanatory variables, SALARY and TSCORE, have small elements. In contrast,

some of the elements for the other regressors are more substantial. This plot does a

fair job of highlighting regressors, but a poor job of highlighting observations.

Figure 3.11 is the profile plot of Ivmaxl. It is easy to see that cases 3 and 18

have multiple regressor values that are influential. In addition, one of the values for

case 2 is influential. However, is to difficult to keep track of which line represents

which column. Another option is to plot "profiles" of each row, as in Figure 3.12.

Here, some of the elements of observations 2, 3, and 18 are highlighted.























5 10 15 20 25 30 35
5 10 15 20 25 30 35


(a) IVmaxl for a2


5 10 15 20 25 30 35


(b) IVm.. for 0


5 10 15 20 25 30 35


(c) Iv21 for f


Figure 3.8: Hills data; X + W pert.; column profiles of directional vectors. DIS-
TANCE values are connected by solid line. CLIMB values are connected by a dashed
line.


























5 10 15 20 25 30 35


(a) IV|max for &2


5 10 15 20 25 30 35


(b) IVmaxl for 3


5

5
5 10 15 20 25 30 35
5
5

(
1

(C) IV21 for


Figure 3.9: Hills data; Model II; X+ W pert.; column profiles for directional vectors.
DISTANCE values are connected by solid line. CLIMB-RATE values are connected
by a dashed line.



















* a


. "" 2Q 40.
*


* *
- ~ .- -a


*.6Q 80
* a


Figure 3.10: School data; X + W pert.; Vmax for directed influence on f3









0.4



0.3



0'.2


i~~ ~ ,J/ '






5 10 15 20

Figure 3.11: School data; X+W pert.; Column profiles of VmaxI for directed influence
on /. Thick solid line- SALARY, thin solid line- WCOLR, dot-dashed line- SES,
small-dashed line- TSCORE, large-dashed line- MOMED.


0.4[


0.2






-0.2



-0.4


* 160
























2 3 4 5

Figure 3.12: School data; X + W pert.; row profiles of IVmaxl for directed influence
on /3 with large elements labeled.


A bubble plot of I Wmax I provides the best graphical display of the information
in Vma, (Figure 3.13). Each element is represented by a point plotted according to
both the corresponding regressor (x-axis) and case (y-axis). Around each point is a
bubble with radius equal to the absolute value of the element, and so large bubbles
indicate large contributors to the direction of maximum curvature. Again, we see

that the elements corresponding to x18is,2, x18is,5, x3,2, x3,5, and x2,3 have the largest

influence, as evidenced by having the largest bubbles.







86












Row

20





0 0

0

0 a


C.
15 0.

10 *

*0*~




*
10 .







0 0 0

0 *












S1 2 3 4 5 Column

Figure 3.13: School data; X + W pert.; bubble plot of IWmax for 3













CHAPTER 4
LOCAL ASSESSMENT FOR OPTIMAL ESTIMATING FUNCTIONS

It is possible to examine the local influence of perturbations on measures other
than the likelihood displacement, LD(w), and the profile likelihood displacement,

LD[G1](w). Several authors have applied local influence techniques to specific
measures of influence. For instance, Lawrance (1988) and Wu and Luo (1993c)
looked at influence on the Box-Cox transformation parameter. Also, influence
has been examined on the residual sums of squares in regression (Wu and Luo
1993b), deviance residuals and confidence regions in GLMs (Thomas and Cook 1990;

Thomas 1990), and goodness-of-link test statistics in GLMs (Lee and Zhao 1997).
However, considerably less attention has been given to assessing local influence
when estimation is not performed by ML. Some work has been done on extentions

to Bayesian analysis (McCulloch 1989; Lavine 1992; Pan et al. 1996), and Lp norm

estimation (Schall and Gonin 1991).
In this chapter, local influence techniques are extended in two ways. First,

we consider the influence of perturbations on a general class of influence measures.

Second, we address parameter estimates that are found via estimating equations.
The intent is to unify nearly all previous work on this methodology and provide a

conceptual basis and computational tools for future applications. The first three
sections outline the inferential platform, a general class of influence measures,

and assumptions about the perturbations. Next, techniques for local assessment

of perturbations are discussed. In Section 4.4, we give some algebraic results
for the two main diagnostic quantities: the gradient vector and the acceleration
matrix. Section 4.5 contains computational formulas, while Sections 4.6-4.8 contain
applications. Several measures, including LD(w) and LD['] (w), are considered in








section 4.6 using ML. Section 4.7 addresses influence on the generalized variance
with an emphasis on ML. Finally, the last section considers quasi-likelihood.

4.1 Inferential Platform

This section outlines assumptions about estimation, the nature of
perturbations, and how the effects of perturbation are measured.

4.1.1 Estimating Functions

Suppose that a statistical analysis is performed to estimate p parameters,
0. Further, suppose that point estimates, 0, are obtained as solutions to a set of p
unbiased estimating equations,

h(O; y) = 0, (4.1)

where for each estimating function, hi(O; y),

Eo(hi(O;y)) =0 i= 1,...p.

Sometimes these estimating equations arise from minimizing or maximizing some
criterion with respect to 0. Examples include least-squares normal equations, score
equations and equations resulting from minimizing a Bayes risk. Quasi-likelihood

(QL) equations (Wedderburn 1974) were originally motivated by maximizing a
quasi-likelihood function, but the existence of the objective function is not necessary.

Generalized estimating equations (GEEs) (Liang and Zeger 1986), like QL equations,
merely require assumptions about first and second moments.
The theory of estimating equations (EEs) dates back to early work by V. P.
Godambe (Godambe 1960; Godambe 1976). There has been renewed interest in the
subject due to new estimation techniques for semi-parametric models and models for
clustered non-normal responses (Zeger and Liang 1986; Prentice 1988; McCullagh
and Nelder 1989; Desmond 1997a). A considerable amount of work has centered








on obtaining optimal estimating functions (OEFs) (Godambe 1976; Crowder 1986;

Crowder 1987; Godambe and Heyde 1987; Firth 1987; Godambe and Thompson
1989; Godambe 1991). To this end, the efficiency matrix of a set of p estimating
functions is defined as

Eff(h) = (E [ ] )-1V[h](E h (4.2)

Functions that minimize this non-negative definate matrix are considered optimal.
Roughly speaking, this index of efficiency is small when the variance of the
estimating functions is small and the average gradient is large. Provided that the
estimating equations arise from maximizing or minimizing some function, we may
interpret that large gradients indicate the function is not flat around its extremum.
Assuming that a fully parameteric form for the data is assumed, the score
functions are the unique OEFs up to a scalar multiple (Godambe 1960; Bhapkar
1972). However, with fewer assumptions about the data, it is generally not possible
to find a unique set of optimal estimating functions. Instead, optimality is bestowed
upon estimating functions within a certain class. One such class of estimating
functions are those that are linear functions of the data.

Under mild regularity conditions, the asymptotic variance of 0 is in fact given
by (4.2) (Morton 1981; McCullagh 1983; Crowder 1986). Thus, minimizing Eff(h)
is equivalent to minimizing the asymptotic variance of the parameter estimates.
Note that the first interpretation of the efficiency matrix is appealing in that it is a
finite sample index, but somewhat awkward in that it is based on a property of the
estimating functions. The second interpretation is appealing in that it is based on a

property of the estimates, but limited in that it requires asymptotics.
For the purposes of applying local influence diagnostics we assume only that
the estimating functions have been standardized. That is, given a set of unbiased
estimating functions, h*(0; y), we utilize the standardized estimating functions given








by

h(O;y) =-E [-I-O (V [h*])-lh'. (4.3)

This standardization has three implications. First, V(h(0; y))-1 is equivalent
to (4.2) and hence equivalent to the asymptotic variance of the parameter estimates.
In addition, the standardized estimating functions are optimal in the following sense.
Given a set of unbiased estimating functions, {h(O; y)}, the OEFs amongst the
class of linear combinations of the h* are given by (4.3). Hence, the standardized
estimating functions are an optimal rescaling of the original estimating functions.
Note that the negative sign is present in (4.3) so that the score equations, rather
than the negative of the score equation, are considered standardized. Finally,
standardized estimating functions are "information unbiased" (Lindsay 1982),
meaning that the following equality holds:

-E [Oh = E[hh'] = V[h]. (4.4)

Hence, V[0] = -E[Oh/0O']. This fact is used later in the chapter to derive influence
diagnostics for confidence ellipsoids.

4.1.2 Perturbations
Perturbations are again introduced as a vector of q real numbers, Wo. These
perturbations may affect the data, the model and its assumptions, or perhaps even
just the estimation technique itself. Examples of data perturbations are minor
changes or rounding errors in the response and the explanatory variables. An
example of model perturbation is the variance perturbation scheme used in Chapters
1 and 2. Finally, perturbing the contribution of each observation to the estimating
functions by case-weights is an example of perturbing the estimation technique.
This perturbation does not affect the distributional assumptions of the data, but
rather how each observation is utilized for inference. Another perturbation to the








inference method would be to allow the value of the hyper-parameters in a Bayesian
analysis to vary. Though some might argue that the hyper-parameters are part of
the model, their values are often chosen for computational convenience.
For the purposes of this chapter, we assume that the perturbation scheme

results in a set of p perturbed estimating equations,

h,(0; y) = 0, (4.5)

which have solutions 6,. It is also assumed that there exists a null perturbation, wo,

whose estimating equations are identical to the original ones. Finally, it is assumed

that there are no constraints on w that would limit their values in a neighborhood of
w0. For example, constraining the space of variance perturbations to positive values

is allowed. However, constraining the perturbations to sum to one is not allowed.
These kinds of constraints could be accommodated, but are not considered here.

4.1.3 Influence Measures

The final item to specify is a quantity whose sensitivity to perturbation is of
interest. This influence measure serves as a barometer for how perturbation effects

inference. The measure can depend on w directly and indirectly via some r x 1

vector of functions of the perturbed parameter estimates, -y(O0,). Some influence
measures, m(w, ,(O)), may be appropriate regardless of the method of estimation

and the perturbation, while others may be more application-specific.
Examples of rather general objective measures are the following:

ml = x'16 (4.6)

M= (4.7)

m3 (Y (48)







where Ew(yi) = p. None of these three measures depend on w directly. The first is
a linear combination of the perturbed parameters. Here, 7(0O,) = z'O,, is a scalar.
The second measure is the ratio of the ith unperturbed parameter estimate to the ith
perturbed estimate. Again, y(9w) = 6L is a scalar. The third measure resembles a
Pearson goodness-of-fit statistic based on the perturbed estimates. Since it is not yet
specified how the means, fA,, depend on the estimates, we simply write y(0b,) = w.
The following measures are more application-specific, in that they depend on
how inference is performed and perhaps even upon the perturbation itself:

m4 = LD(w) = 2[L(0; y) L(0,; y)] (4.9)

m5 = LD'\(w) = 2[L(01,62; y) L(01,, g(10,);y)] (4.10)

m6 = LD*(w) = 2[L,(6,; y) L0,(; y)] (4.11)


f(y I 0) ]
m7 = E [1og(f/(Y I 00) (4.12)


m8 = log (h(O ;Y) (4.13)

The first of these measures is the likelihood displacement, for which "Y(0,) = 0b.
The second is the profile likelihood displacement, which has -Y(6O)' = ( g(06i)'),
where g(01.) maximizes the profile likelihood of 01, with respect to 02. The third
measure is a likelihood displacement with a "moving frame" because the base
likelihood of the measure is the perturbed one. Obviously, these three measures are
intended for use when estimation is performed by maximum likelihood.
The influence measure m7 is a Kullback-Leibler divergence between the
densities f(y I 0) and f(y | 0I). The quantity is non-negative, achieving its
minimum value of zero when the two densities are equivalent. Here, Y'(0w) = 0Ow.
The last measure addresses sensitivity of asymptotic standard errors. Recall
that the determinant of a matrix equals the product of its eigenvalues, and when
the matrix is a variance-covariance matrix for a vector of estimates 0, the squared




Full Text
66
Proceeding in a similar fashion to the derivations for perturbing a single column,
partial derivatives with respect to u are as follows:
dLu_ J_
du) o2
fa(y X/3) faW/3
fa(y-Xfa-faW0
fa(y-X/3)-faW/3
(3.19)
Since the null perturbation is 0, evaluating at u0 simplifies the expression:
Mv X0)
Mv x$)
Mv X0) _
We then proceed to take second partial derivatives and evaluate at 9:
o
o
( fax N
&K(e-.y)
1
0 r 0 ... 0
fax
dud fa
9=90,u=u0
: :
\
0 r)
^ fax j
----- ^(1* <8> r ¡3 X)
dLu{9\ y)
du
U=U0
&*LU(0\V)
dudo2
9=90,U=U0
far
far
far
= *)



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PAGE 212

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PAGE 213

/' 0 + 81,9(56,7< 2) )/25,'$


88
section 4.6 using ML. Section 4.7 addresses influence on the generalized variance
with an emphasis on ML. Finally, the last section considers quasi-likelihood.
4.1 Inferential Platform
This section outlines assumptions about estimation, the nature of
perturbations, and how the effects of perturbation are measured.
4.1.1 Estimating Functions
Suppose that a statistical analysis is performed to estimate p parameters,
0. Further, suppose that point estimates, 0, are obtained as solutions to a set of p
unbiased estimating equations,
h(0,y) = 0, (4.1)
where for each estimating function, /(0; y),
E9(hi{0;y)) = 0 i = l,...p.
Sometimes these estimating equations arise from minimizing or maximizing some
criterion with respect to 6. Examples include least-squares normal equations, score
equations and equations resulting from minimizing a Bayes risk. Quasi-likelihood
(QL) equations (Wedderburn 1974) were originally motivated by maximizing a
quasi-likelihood function, but the existence of the objective function is not necessary.
Generalized estimating equations (GEEs) (Liang and Zeger 1986), like QL equations,
merely require assumptions about first and second moments.
The theory of estimating equations (EEs) dates back to early work by V. P.
Godambe (Godambe 1960; Godambe 1976). There has been renewed interest in the
subject due to new estimation techniques for semi-parametric models and models for
clustered non-normal responses (Zeger and Liang 1986; Prentice 1988; McCullagh
and Nelder 1989; Desmond 1997a). A considerable amount of work has centered


126
4.8.1 Quasi-likelihood
Consider a model in which a set of n independent observations y depends
on a set of p parameters 0 through their means n(0). Although full distributional
assumptions are not made, we also assume that the variances of y depend on their
means: V[y] = cr2V(n). A class of estimating functions that we may consider for
estimating 6 are given by
h'(e-,y) = W(0)(y-n(e))/c2, (4.71)
where the p x n weight matrix W (0) can be a function of the unknown parameters.
Standardizing (4.71) as in (4.3) yields (Wedderburn 1974; Godambe and Heyde
1987):
h{0] y) = %-V(v)~1(y V{0))/o2- (4-72)
These optimal estimating functions are called the quasi-score functions because
the solutions to h(0,y) = 0 maximize the following quasi-likelihood function,
provided the integral exists:
Q = [^T)dt (473)
4.8.2 Local Assessment
Some preliminary results for local assessment of quasi-likelihood models for
independent data are presented in this section. Assumptions about the models
form are presented, some perturbation schemes are proposed, and a case-weight
perturbation is applied to an example data set.
4.8.2.1 Assumptions
It is assumed that observations are independent, and that the variance is
related to the mean by the function V(pi). In addition, it is assumed that the mean


200
Thomas, W. (1991), Influence Diagnostics for the Cross-validated Smoothing
Parameter in Spline Smoothing, Journal of the American Statistical
Association, 86, 693-698.
Thomas, W. and Cook, R. D. (1989), Assessing Influence on Regression
Coefficients in Generalized Linear Models, Biometrika, 76, 741-749.
Thomas, W. and Cook, R. D. (1990), Assessing Influence on Predictions From
Generalized Linear Models, Technometrics, 32, 59-65.
Tsai, C.-L. and Wu, X. (1992), Assessing Local Influence in Linear Regression
Models With First-order Autoregressive Or Heteroscedastic Error Structure,
Statistics & Probability Letters, 14, 247-252.
Vos, P. W. (1991), A Geometric Approach to Detecting Influential Cases, The
Annals of Statistics, 19, 1570-1581.
Wang, S.-J. and Lee, S.-Y. (1996), Sensitivity Analysis of Structural Equation
Models With Equality Functional Constraints, Computational Statistics and
Data Analysis, 23, 239-256.
Wang, X., Ren, S., and Shi, L. (1996), Local Influence in Discriminant Analysis,
Journal of Systems Science and Mathematical Science, 8, 27-36.
Wedderburn, R. W. M. (1974), Quasi-likelihood Functions, Generalized Linear
Models, and the Gauss-Newton Method, Biometrika, 61, 439-447.
Weiss, R. (1996), An Approach to Bayesian Sensitivity Analysis, Journal of the
Royal Statistical Society, Series B, 58, 739-750.
Weissfeld, L. A. (1990), Influence Diagnostics for the Proportional Hazards
Model, Statistics & Probability Letters, 10, 411-417.
Weissfeld, L. A. and Schneider, H. (1990), Influence Diagnostics for the Weibull
Model Fit to Censored Data, Statistics & Probability Letters, 9, 67-73.
Wu, X. and Luo, Z. (1993a), Residual Sum of Squares and Multiple Potential,
Diagnostics By a Second Order Local Approach, Statistics & Probability
Letters, 16, 289-296.
Wu, X. and Luo, Z. (1993b), Residual Sum of Squares and Multiple Potential,
Diagnostics By a Second Order Local Approach, Statistics & Probability
Letters, 16, 289-296.
Wu, X. and Luo, Z. (1993c), Second-Order Approach to Local Influence, Journal
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Wu, X. and Wan, F. (1994), A Perturbation Scheme for Nonlinear Models,
Statistics & Probability Letters, 20, 197-202.


109
4.5.1.1 Remarks
The formula for P can be combined with the results of Theorem 4.2 to
further revise the acceleration matrix:
d2m(u),'y(0UJ))
M =
dujduj1
I J'K'
I J'K'
I
KJ
I
KJ
+ [*;<8> J'] GJ+[s\KIq]P
+ [; j'} gj
+ [s'^K Iq\
1
J
-ft [ J, J' ]
-
J
I J'K'
+
I
KJ
+ [; j'] gj
. 1
I
s\Kh [ lq J' }

J
(4.44)
Also, we may write
p
[s'^K Iq]P = Y^(s'iKi)Pi
i=l
P
= £(-*>)
3=1
=1
p p
j=1 =1
/9 J'
Iq J'
*3
*9
J
£ W-*;
3-1
J, J'
(4.45)


136
E(ti|y) = E(w) is Z/3, and its mean squared error of prediction can be shown to be
a2(Z'(X'X)~1Z + /m). Also, Z(3 is normally distributed. These facts can be used
to create a 100(1 a)% prediction interval for a single u with covariate vector z\
z'P Za/2a2(z'(X'X)~lz -f 1). (5.3)
In the event that a2 is unknown, the prediction interval given in (5.3) is
still valid but cannot be calculated. One option is to simply replace <72 by s2 to
produce an approximate prediction interval. A better alternative is to make use of
the following pivotal quantity:
u z'¡3
'-nki
y/s2(z'(X'X)~1Z + 1)
where k is the number of regressors. An exact 100(1 a)% prediction interval is
then
(5.4)
z'0 /2^sHz'(X'X)-'z + 1).
(5.5)
This simple situation highlights the fact that unknown parameters can
complicate the prediction problem. While pivotal quantities may be available in
simple situations, accomodating unknown parameters may be more cumbersome
in complicated models. Therefore, attempts have been made to unify predictive
inference via predictive likelihood.
5.2 Predictive Likelihood
Predictive likelihood is a likelihood-based framework for handling the
problem of prediction. The goal is to predict u via a point or interval predictor in
the presence of unknown parameters 6. If u is treated as a vector of parameters,
then the joint likelihood of u and d is
L'(u, 8, y) = f(y, u \ 8) = f,(u | 8, y)f2(y | 8).
(5.6)


38
2.6.2.1 1-to-l coordinate changes in ft
Following Loynes (1986) and Beckman et al. (1987), let us consider 1-to-l
coordinate changes in the perturbation space 2. That is, a new perturbation scheme
is constructed with elements = k(ui) for i 1,... q, for a smooth function k. The
new schemes perturbation, uj*, will not yield the same curvature as a;. Specifically,
Cu* = (§£:)w_ Cu, where uio denotes the null perturbation value for w*.
2 2
As an example, consider perturbing the variance of yi to be % rather than 7-.
Here, k(uji) uj?, resulting in new curvatures that are (§^U;=i)2 = (2u;)2.=1 = 4
times the original ones. Hence, despite the fact that the two versions of 7 can
produce the same set of perturbed models, the curvature in identical directions will
change. This implies that a universal benchmark for curvatures is not possible.
Indeed, given the arbitrariness of scaling the perturbation, it should come as no
surprise that curvatures cannot be compared across perturbation schemes. However,
the direction of maximum curvature is unchanged under 1-to-l coordinate changes
in 2.
2.6.2.2 Interpreting curvatures
In this section, a simple way to assess curvatures from the same perturbation
scheme is given. First, y(7 is a Taylor series approximation to LD(u)0 + av)
(Lawrance 1991; Escobar and Meeker 1992). Also, assuming that n is large, an
asymptotic 100(1 a)% likelihood-based confidence region for 0 is given by values
of 9 such that
2(L(0;y) L(Q-y)) < xl,i-a-
Thus, if twice the curvature in direction v is larger than xl,i-Q, then a perturbation
at a distance of one from u?o in direction v moves the parameter estimates to the
edge of the asymptotic confidence region.


79
Example 1
Consider the Scottish hills data set described in Chapter 1. The purpose of
this example is twofold- to demonstrate how the X-matrix results can be displayed
graphically, and to demonstrate how collinearity affects the curvature values when
examining influence on $.
Figure 3.7 gives the direction of maximum curvature for directed influence on
a2. The first 35 elements correspond to perturbations of the DISTANCE regressor
values, while elements 36-70 correspond to perturbations of the CLIMB regressor
values. The value for race 18s DISTANCE is the largest contributer to this
influential perturbation. Elements 7 (DISTANCE for race 7) and 53 (CLIMB for
race 18) are also large
The information in Figure 3.7 can be difficult to discern because the
perturbation vector enters into the likelihood in the form of a matrix, W. Another
way to display umax is to first take its absolute value and then to consider the
elements corresponding to each column of W. Each portion can then be plotted
against indices 1,..., n and shown together (Figure 3.8(a)). This graphical display
provides a profile plot of each column of Ws contribution to the direction of
maximum curvature. Here we see again that elements corresponding to observations
18 and 7 as being the most influential. There is also some evidence that measurement
error in the DISTANCE values (solid line) are more influential than error in the
CLIMB values (dashed line).
Figures 3.8(b) and 3.8(b) provide profile plots of the absolute value of the
.. r two directional vectors corresponding to V\ and of F The first plot draws
attention to observations 7, 11, and 18, while no observations appear noteworthy in
the second plot.
At this point we turn our attention to the curvature values that accompany
these directional vectors (Table 3.3). The first column shows the curvatures for local


28
2.4.1 Building Blocks
The perturbed likelihood is
77
Lu(0; V) oc -- loga2 ~ X/3 + u)\y X0 + u)
= log o'2 ^(y + w)'(y + w)
Tt 1
= --loga2 2^2(y'y + 2w'y + w'w),
where y = y X(3. The next step is construction of A. Partial derivatives with
respect to u are as follows:
d Lu(9\y)
du
~(y-x¡8 + ).
Second partial derivatives are then
dud ¡3'
d2Lu{0\y)
duda2
= -tX
= -7(y X^ + u;).
Evaluating at the original model yields
&Lu{9-,y)
dud/3'
d2Lu(0;y)
= X
G=0,u=uo
1
= T-rV.
G=6,u=u0
duda2
From these quantities, the n x p matrix A' can be constructed:
1
A' =
X hr


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
LOCAL ASSESSMENT OF PERTURBATIONS
By
Glen Lawson Hartless
August 2000
Chairman: James G. Booth
Major Department: Statistics
Statistical models are useful and convenient tools for describing data.
However, models can be adversely effected by mistaken information, poor
assumptions, and small subsets of influential data. Seemingly inconsequential
changes to the observed data, the postulated model, or the inference technique may
greatly affect the scientific conclusions made from model-fitting. This dissertation
considers local assessment of perturbations in parametric and semi-parametric
models. The effects of perturbation are assessed using a Taylor-series approximation
in the vicinity of the fitted model, leading to graphical diagnostics that allow the
analyst to identify sensitive parameter estimates and sources of undue influence.
First, computational formulas are given to assess the influence of perturbations
on linear combinations of the parameter estimates for maximum likelihood. A
diagnostic plot that identifies sensitive fitted values and self-predicting observations
in linear regression is introduced. Second, the eigensystem of the acceleration matrix
for perturbing each element of the design matrix in linear regression is derived.
vii


67
From these quantities we can construct the np x p matrix A', as given by Cook
(1986):
Ikr-j3X
(3.21)
where <8> denotes the Kronecker product (Searle 1982).
The next step is to construct the acceleration matrix. Here we use the fact
that (A (g> B)(C D) AC BD, assuming comformability of the matrices.
Noting that A = A\ = \ A, the acceleration matrix can be written as
F = (/* r p 0 x) (X X)~l (jfc r' X') + (90' rr')
= ^ (ijk <2) r P x) ((X'X)1 (8> r' P (X'XJ^X')
(3.22)
3.4.2 Directed Influence on d2
The acceleration matrix for the profile likelihood displacement for second term of (3.22):
4
nu4
/3P <8> rr'.
(3.23)
This matrix has a single non-zero eigenvalue, -^-||/3||2, with corresponding
eigenvector oc (3 r. This solution is confirmed below:
(F1* j AI)v oc
43p'rr'-^\\p\\2I ($r)
no
o
no*
no4
PP' rr' -\\P\\2\\r\\2I (¡3 r)
\\P\\2P IM|2r ||/9||2||r||2(j9 r)
= 0.
(3.24)


77
Proof of Theorem 3.2 We begin by obtaining update formulas for A and L 1 as
per Section 2.7:
=
= 0X) = r ¡3 Xip)
(apt spy'
\ da ' da')
(i-y(xx))
(H 1
a2D(l/S).
(3.36)
(3.37)


bs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
189
Dep Var
Predict
Std Err
Std Err
Student
VSCORE
Value
Predict
Residual
Residual
Residual
0.3314
0.2714
0.227
0.0600
0.260
0.231
-1.4737
-1.4135
0.228
-0.0602
0.259
-0.233
0.2454
0.9244
0.099
-0.6790
0.330
-2.058
0.9657
1.0471
0.120
-0.0814
0.323
-0.252
0.3468
0.2126
0.123
0.1342
0.322
0.417
-0.2033
-0.1885
0.231
-0.0147
0.256
-0.058
1.1548
1.0312
0.150
0.1236
0.310
0.398
-0.2892
-0.2146
0.083
-0.0747
0.334
-0.223
1.0190
0.9116
0.167
0.1073
0.301
0.356
0.3640
0.3278
0.260
0.0362
0.226
0.160
-2.0255
-1.6459
0.169
-0.3796
0.300
-1.265
0.0202
-0.2800
0.205
0.3002
0.277
1.083
-0.0314
0.1489
0.195
-0.1803
0.284
-0.634
-0.3408
-0.2814
0.084
-0.0594
0.334
-0.178
-2.1287
-1.8229
0.188
-0.3058
0.289
-1.058
0.7938
0.5707
0.113
0.2230
0.326
0.685
-0.5643
-0.3168
0.169
-0.2475
0.300
-0.825
-0.5815
-1.4413
0.181
0.8598
0.293
2.932
1.3783
1.1853
0.167
0.1930
0.301
0.640
1.0190
0.9738
0.143
0.0452
0.313
0.144
Cooks
Hat Diag
Cov
-2-1-0 1
2
D Rstudent
H
Ratio
Dffits
1
1
0.008
0.2236
0.4325
2.4440
0.1952
1
1
0.008
-0.2251
0.4362
2.4596
-0.1980
**** |
1
0.077
-2.3475
0.0831
0.2929
-0.7067
1
1
0.002
-0.2441
0.1212
1.5729
-0.0906
1
1
0.005
0.4055
0.1281
1.5276
0.1555
1
1
0.001
-0.0557
0.4499
2.5639
-0.0504
1
1
0.007
0.3868
0.1891
1.6511
0.1868
1
1
0.001
-0.2160
0.0575
1.4734
-0.0534
1
1
0.008
0.3457
0.2354
1.7697
0.1918
1
1
0.007
0.1545
0.5680
3.2410
0.1771
** 1
1
0.102
-1.2931
0.2414
1.0588
-0.7295
1 **
1
0.128
1.0900
0.3527
1.4516
0.8046
* |
1
0.038
-0.6212
0.3193
1.8106
-0.4254
1
1
0.000
-0.1720
0.0588
1.4844
-0.0430
** 1
1
0.094
-1.0629
0.2964
1.3615
-0.6899
| *
1
0.011
0.6725
0.1069
1.3486
0.2326
* |
1
0.043
-0.8152
0.2406
1.4745
-0.4589
1***** |
0.654
4.3370
0.2756
0.0276
2.6748
1 *
1
0.025
0.6272
0.2346
1.6059
0.3472
1
1
0.001
0.1394
0.1727
1.6948
0.0637


4.5.0.1 Computation of J
The following theorem generalizes equality (14) in Cook (1986).
104
Theorem 4.3
Let 6u be the solutions to the p perturbed estimating equations y) = 0. Then,
J = h 1 A, (4.33)
where
A =
dhu(0;y)
du'
(4.34)
Proof of Theorem 4.3 By assumption, we have the following equality
hu(du;y) = 0.
Taking derivatives with respect to u' on both sides we find
dhu(0;y)
du'
+ dhu(0u\y) d0 = Q
0=e d' du' ~
(4.35)
where 0 is now a p x q matrix. Evaluating both sides at the original model we find
A + hJ 0,
and the theorem follows directly.
Theorem 4.3 has two implications on computation: the gradient vector
is easy to calculate, and the acceleration matrix is easy to calculate provided
sy = 0. Neither h nor A require that the solutions to the estimating equations
be closed-form for either the perturbed or the unperturbed inferences. In addition,
the matrix h is often computed as part of the estimation process. Not only is
it a by-product of using the Newton-Raphson root-finding technique to solve the
estimating equations, but it serves as an estimate of the asymptotic variance of 0,
the inverse of (4.4). When estimation is performed by ML, h 1 is the inverse of


60
(a)
^max for ¡31
l
0.75
0.5
0.25
0.25
0.5
0.75
1

5 10 151 120

(b)^max for
1
0.75
0.5
0.25
0.25
0.5,
0.75
-1

. >1. v 15 *
(c)^max for $3
(d)^max for /?4
1
0.75
0.5
0.25
-0.25
0.5
0.75
1
. ** *
5 V 15 V
1
0.75
0.5
0.25
-0.25
0.5
0.75
1
' l
* *5 *10 *15 20
(e) ^max for /?5
(f) Umax for a2
Figure 3.3: School data; X3 + u> perturbation; directed influence plots


187
The foundation of the local influence methodology is a Taylor series
approximation to a high-dimensional combined with some familiar geometry
concepts: slopes and accelerations. Using these ideas, the generalization to a class of
influence measures for optimal estimating functions in Chapter 4 is straightforward.
This extention of the methodology provides many avenues for future research;
quasi-likelihood and mixed model estimation are only the beginning of the areas of
application.
As for other areas of research, the idea of perturbations is an excellent
concept for linking influence diagnostics, sensitivity analysis, robust inference,
and aspects of optimal design. Robust estimation techniques are those that yield
inferences that are largely insensitive to perturbations. Consequently, minor aspects
of the data, model, and inference technique do not overly influence scientific
conclusions. If a perturbation and influence measure are appropriate for several
methods of inference, the maximum local influence could be used to compare the
techniques robustness. In fact, mimimizing the maximum local influence could even
serve as a criterion for robustness (minimax li estimation). Optimal sampling and
experimental designs, which are efficient under assumed conditions, should also be
designed so that they are insensitive to perturbation. Local assessment could be
used to assess how sampling and experimental designs can ameliorate or compound
influential phenomenon.


153
where c = A BD 1C. Applying this result, we find
-l
-h =
a
T
1
c
_D(£!5i2i)-in D(^^i)-1nn'D(^^)-1 + A:D(^|^)-1
<7
T
al + i
J
err
^2+:
(5.35)
where
rij) =
(*2+£)(*f+£)
(-c2+^)(-2+^)
k = n~Y_]
nial
K2+^)(-?+^)
y rr2
- = V -
' /T2 -I- /7
and
(T? + a2/rii ri o2c + a2/rii
i=i 0 i=i '
2
When the design is balanced, k = ^^2^1 and the following simplifications can be
made:
-1
-h = (ac/g + a /n)
~ffc!? 1'
a%/g+cr2/n 9
2
,,-^/g x [._# ) i k *2/"
a2 / g+a2 / n 9 \cr2/g+cr2/n J i9x9 K a2/g+cr2 /n
l/9 + ^/n
-^1
3
.1'
9 9
Hjl I i 2_ r
p2 -*-

26
2.3.4 Acceleration Matrix for Directed Influence on f3
When directing influence on we have
L 1 B22 =
-d2(XX'Yl 0
0' 0
The resulting acceleration matrix is
F1"1 = 22A (XX')-' Ag.
(2.6)
Note that this implies that directed local influence on /3 is equivalent to local
influence for when treating a2 as known and equal to its MLE.
2.3.5 Acceleration Matrix for Directed Influence on a Single Regression Parameter
By applying the general results of Beckman et al. (1987), we can construct T
(2.4) for directed influence on in a regression model. Using results for the inverse
of a partitioned matrix, TT' = (L 1 1322) can be expressed as
d
1
0
-x;x(1)(x;1)xM)-> 0
ot_,
011 0
5
(2.7)
where d= X[X, X^X^X^X^-'X^Xi.
We denote (X|1)X(i))_1X|1jX1 by qq, as it is the regression parameter
estimates from modeling X\ as a linear combination of the other explanatory
variables. This simplifies the notation of (2.7) greatly:
~(L 1-B22)
1 -7 0
-7i 7i7 0
0 0' 0
(2.8)


where
31
Z = Xcp
a = ip'(3
ipi = the ith eigenvector of X'X.
Because model. The new regressors, Zx = Xtpx, ...,Zk = X(pk, are known as the principal
components, and they are mutually orthogonal. In addition, each satisfies the
characteristic equations of F
(2.14)
= 0
Therefore, the standardized principal component regressors can serve as an
orthogonal set of directional vectors, each of which has maximum curvature. Note
also that the k adjusted regressors, rx,...,rk, are also directions of maximum
curvature, but they do not constitute an orthogonal set.
2.4.3.2 Influence on 6
The acceleration matrix F given in (2.10) has two non-zero eigenvalues:
and js, with the second one having multiplicity k. The first eigenvector is oc r,
and the set of k principal components can be used to complete an orthogonal set of
eigenvectors. Comfirmation of these solutions is straightforward.
2.5 Unequal Measurement Precision Applied to Regression
Cook introduced this perturbation in his original (1986) paper on local
influence, and the results in this section come from his work. Annxl vector
of perturbations is introduced such that the perturbed likelihood is based upon


168
Appendix B contains model-fitting information from an analysis in which the
the observation is included. Note that the regressors are also centered and scaled
for this analysis. Additionally, the variance components have been estimated by
restricted maximum likelihood (REML). By not perturbing the REML estimating
functions, the local influence techniques described in this chapter can be used to
assess the effects of perturbations on ¡3 and while effectively treating the variance
components as known.
The goal of the analysis is to first establish a linear relationship between
the satellite data, which is readily available, and the actual reported data, which
is accurate but not available for all segments in all counties. The resulting model
can then be used to make predictions for segments in which reported hectares are
unavailable.
Interest centers on predicting the population mean hectares of corn per
segment. Because the satellite data are available for all segments in each of the 12
counties, the population mean number of pixels for corn and soy in county i is known.
These means are denoted Xu(p) and x2(p) and respectively. The corresponding
population mean hectares of corn per segment in county i is then defined as
Vi 00 T Ui T Aorn^lt(p) T (3soyX2i(py
Point inferences for these values can be made using the the corresponding BLUPs:
Vi 00 T V>i + orn^li(p) T PsoyX2i(p')
5.9.2 Residual Analysis
Figure 5.1 contains diagnostic plots of the corn data. The first plot is of the
37 composite residuals c = y X0. The three single-observation counties appear
first, followed by the county with two obervations, etc. Cases from the same county
(cluster) are joined by a line, a practice suggested by Hilden-Minton (1995). This


197
Kim, M. G. (1995), Local Influence in Multivariate Regression, Communications
in Statistics, Part A Theory and Methods, 24, 1271-1278.
Kim, M. G. (1996b), Local Influence in Box-Cox Transformation for Multivariate
Regression Data, Communications in Statistics, Part A Theory and
Methods, 25, 2955-2969.
Kim, M. G. (1996c), Local Influence in Multivariate Normal Data, Journal of
Applied Statistics, 23, 535-541.
Kim, M. G. (1998), Local Influence on a Test of Linear Hypothesis in Multiple
Regression Model, Journal of Applied Statistics, 25, 145-152.
Kim, Y.-I. (1991), Local Influence Approach Diagnostics for Optimal
Experimental Design, Korean Journal of Applied Statistics, 4, 195-207.
Lavine, M. (1992), Local Predictive Influence in Bayesian Linear Models With
Conjugate Priors, Communications in Statistics, Part B Simulation and
Computation, 21, 269-283.
Lawrance, A. J. (1988), Regression Transformation Diagnostics Using Local
Influence, Journal of the American Statistical Association, 83, 1067-1072.
Lawrance, A. J. (1991), Local and Deletion Influence, In Stahel, W. and
Weisberg, S., editors, Directions in Robust Statistics and Diagnostics. Part I,
pages 141-157, New York: Springer-Verlag.
Lee, A. H. and Zhao, Y. (1996), Assessing Local Influence in Measurement Error
Models, Biometrical Journal. Journal of Mathematical Methods in
Biosciences, 38, 829-841.
Lee, A. H. and Zhao, Y. (1997), Assessing Influence on the Goodness-of-link in
Generalized Linear Models, Statistics & Probability Letters, 31, 351-358.
Lee, Y. and Nelder, J. A. (1996), Hierarchical Generalized Linear Models (with
discussion), Journal of the Royal Statistical Society, Series B, 58, 619-678.
Lejeune, M. and Faulkenberry, G. D. (1982), A Simple Predictive Density
Function (with discussion), Journal of the American Statistical Association,
77, 654-659.
Lesaffre, E., Asefa, M., and Verbeke, G. (1999), Assessing the Goodness-of-Fit of
the Laird and Ware Model- an Example: The Jimma Infant Survival
Differential Longitudinal Study, Statistics in Medicine, 18, 835-854.
Lesaffre, E. and Verbeke, G. (1998), Local Influence in Linear Mixed Models,
Biometrics, 54, 570-582.


72
parameter estimates are more sensitive to rounding and measurement errors in the
explanatory variables.
Now let us consider vmax as per Theorem 3.1:
i>max a v?fe r p X This vector and the other eigenvectors of Fl provide detailed information about
which elements of X most affect the stability of the MLEs. The expression for
Umax shows how outliers and leverage in the final principal component regressor
determine precisely what kinds of mismeasurements have large local influence on 0.
The directional vector can be used to scale W to achieve maximum local influence:
W
n
(X
ipkir PiX VhkT ~ PkXlfk
(3.31)
The elements in the first column are a weighted average of the residuals (r) and the
regressor values of the last principal component regressor (Xipk). The respective
weights are the first original regressors contribution to the final principal component
( Alternatively, we can think of the elements in the first row as being a
weighted average of the explanatory variables contributions to the final principal
A /
component ( weights are the first residual (ri) and the negative of the first observations value for
the final principal component (x[(pk).
3.4.3.2 A special case
In order to examine in more detail how the perturbations given in (3.31)
affect 0, we consider a simple situation for which there are closed-forms for the
perturbed parameter estimates. Let us assume that the response and a pair of
regressors have been centered and scaled to have length one, and that the regressors


166
Here, the elements corresponding to larger clusters receive more weight. This is in
contrast to the first cluster perturbation scheme, for which umax oc u.
The direction of maximum curvature for i is given by
max OC A fl d2
OC
0 -D(u)
-1
<72 |_ k
tr2+a2/n i ni
(5.52)
(?F++ m
a2-\-a2/n,2 2
)
?!
a2+cr2/ngU9
where k was defined after (5.35). The coefficient for the first element is approximately
1 + g/n\ times larger than the others, indicating that influence on the ith BLUP is
accomplished largely by perturbing the variance of itj. The direction of maximum
curvature for fi + i is even more dominated by the first element:
^max OC A h
-1
1
1
0
oc
i
1
(5.53)
Here, the first element is approximately g 1 times larger than the other elements.


where r1>(J is the residuals from fitting Xi)W as a function of X2,w The building
blocks needed are as follows:
74
= x2,(x'wx2iW)-1x'u =
(Xi-afrXMX't-afoX [)
1 + a? ft
ri,u = (I ~ H2,u)Xi)UJ = (1 afii acp2)Xi + cX2 + ar
r'i,uV = (1 aPi ~ acp2)fii + c/32 + a( 1 - Pi)
Hri^H2 = (1 ap! acP2)2 + c2 + a2( 1 p\ p¡),
where c = a^1~2 l+a2p2
a (1 api acp2)Pi + cP2 + q(1 Pi P2)
(1 aPi acp2)2 + c2 + a2(l Pi Pi)
For the sake of comparison, we can also consider perturbing only the first
column. Perturbation in the direction of umax when directing effects on Pi results in
the following perturbed X matrix:
Xul = X+a*W*
= X+a*
r PiXi 0
(3.34)
a*r + (1 a*Pi)Xi X2
as per (3.8). Here, the perturbed estimate is given by
a (1 a-A)ft + a(l 2 ~ 2)
(1 o-A)2 + a*2(l -b\- 2)'
In order for the two perturbations to be comparable, they should be equal in size.
This is accomplished by stacking the columns of the perturbation matrices and
making the resulting vectors equal length. In other words, the Froebenius norms
of aW and a*W* should be equal. Letting a* =
adjustment.
\A-02
provides the necessary


Table 1.1: Scottish hills race data
Observation
Record Time (min.)
Distance (miles)
Climb (ft)
1
16.083
2.5
650
2
48.350
6.0
2500
3
33.650
6.0
900
4
45.600
7.5
800
5
62.267
8.0
3070
6
73.217
8.0
2866
7
204.617
16.0
7500
8
36.367
6.0
800
9
29.750
5.0
800
10
39.750
6.0
650
11
192.667
28.0
2100
12
43.050
5.0
2000
13
65.000
9.5
2200
14
44.133
6.0
500
15
26.933
4.5
1500
16
72.250
10.0
3000
17
98.417
14.0
2200
18
78.650
3.0
350
19
17.417
4.5
1000
20
32.567
5.5
600
21
15.950
3.0
300
22
27.900
3.5
1500
23
47.633
6.0
2200
24
17.933
2.0
900
25
18.683
3.0
600
26
26.217
4.0
2000
27
34.433
6.0
800
28
28.567
5.0
950
29
50.500
6.5
1750
30
20.950
5.0
500
31
85.583
10.0
4400
32
32.383
6.0
600
33
170.250
18.0
5200
34
28.100
4.5
850
35
159.833
20.0
5000
Source: Atkinson (1986)


121
Letting ai be the ith column of (X'X) it follows that
Sc, = -
xnx[
l(p-l)l
X21X2
a1 -
x21x2
a-
2(p-l)2
xix'n _
Xn2X'n
xn(p-l)xn
p1
na2
xllxi l(p1)1
r -1
£21 2 2(p1)2
ai
p1
XnlXn xn(pl)xn
na
2' s D{X,)X ... D{XP.,)X
'V\-l
vec(X'X)
na
D{Xx) ... D(Xp-i)
[Ip., X] vec(X'X)-1 rsq.
na*
4.7.2.2 Construction of sy
Starting from the Hessian of the unperturbed log-likelihood,
dh(0,y) dL(0; y)
do'
8080'
-X'X -{y-X0)
-My xpy 5*-Mv- *£)'( ~ x&)
(4.65)
the following partial derivatives at the null perturbation can be computed:
d3L(0-y)
dpidpdp
d3my)
da2d/3d/3'
d3L(0;y)
= 0
= ~x'x
= -^X'X
da2d(3da2
d3Wy)
= -RX'(y X/3)
= tsX'r = 0
d{a2Y
=-^+M~xi3Y{v-xi3)
n 3 r'r 2 n
/t6 ST& rr&


1
ACKNOWLEDGMENTS
I would first like to thank my chairman, Jim Booth, and cochairman, Ramon
Littell, for all of their help and guidance. Their combined knowledge, insight, and
mentoring has been extraordinary. I would also like to express my gratitude to them
for allowing me to pursue avenues of research that are probably closer to my heart
than theirs. Thanks also go to the other esteemed professors who served on my
supervisory committee: James Algina, Randy Carter, and James Hobert.
I would also like to acknowledge people who have influenced me academically
and professionally over the years: Chuck Nowalk, John Grant, and many others
in the Phillipsburg and Lopatcong school systems, Walter Pirie and the faculty
at Virginia Tech, Ron DiCola at AT&T/Lucent, Larry Leemis at the College of
William & Mary, Danny Perkins, Bo Beaulieu, and many others at the University of
Florida, and also fellow students Jonathan Hartzel, Phil McGoff, and Galin Jones,
among many others. I would also like to thank Allan West for maintaining an
extremely reliable computing environment.
Thanks also go to my parents for their love and support and for instilling in
me the discipline needed to achieve my goals.
Finally, my utmost thanks go to my wife, Christine. I am forever grateful for
her unequivocal love, unending patience, and thoughtful insight.
IV


Proof of Lemma 4.2 Taking the ith row of (4.35) in the proof of Theorem 4.3
gives the following equality:
106
dhiu(0] y)
du)'
+ dhjujOuiy) 96 w Q/
0- a d' du'
(4.37)
where O' is a 1 x q matrix. Taking derivatives of the first term on the left side of
this equality with respect to u;, we find
9
dhiw(0] y)
doj
9U3'
0=0*.
d2hiu;(0] y)
dijidtjj1
6=0*
+
<90>2M0;y)
du> 909u)'
6=0*
(4.38)
Taking partial derivatives of the second term with respect to cj, we find
9_
du
dhiujO^y) 90 ^
96,, du'
92hiu{0\ y)
9u)90'
+
90 d2hiuJ{0]y)
du3 dOdO'
+E
3=1
6=6*
Z^dhiu(0^y) dOj,
9=9*
OP
du/'
(4.39)
]U1
96jul du)du>'
Using (4.38) and (4.39), the next step is to take derivatives of the equality in (4.37)
with respect to uj and then evaluate the result at the null perturbation. It follows
that
dhiuiO^y) 90 u
( 9
dhi^O-y)
du>'
0=6*
d2hiul(0] y)
+
86,
9u>'
+
J / o
dOl 82hU0-y)
+
dojduj' 0=0u doj dOdu}'
d2hiuJ(d-, y)
6=6*/ o
9uj90'
96u + 96u 92hiw(0\ y)
q=q duj' du: 9090'
6=0.
M
du'
+ E
3=1
dhiui(0u,y) 96
96
du/dic'
(4.40)
= Qi + J'R'i + RJ + J'hiJ +
3=1
9hiu(0; y)
I J'
Qi Ri
R' hi
+
P = 0,
where 0 is now a p x p matrix. The lemma follows immediately.
I
J
96j
hi I a
Pj


CHAPTER 1
INTRODUCTION TO LOCAL INFLUENCE
1.1 Premise
George Box (1982, p. 34) stated all models are wrong; some models are
useful, and A. F. Desmond (1997b, p. 117) stated a robust method ought not to
perform poorly in neighborhoods of the nominally true model. Indeed, conclusions
based on a statistical model should be reliable under small perturbations to the
data, the models assumptions or the particulars of the estimation method. In a
seminal paper, Cook (1986) utilized the basic concept of perturbing the postulated
model or the observed data to develop a variety of diagnostic tools. He showed how
small perturbations can be used to develop easily-computable diagnostics for almost
any model estimated by maximum likelihood (ML) estimation.
1.2 Measuring the Effects of Perturbations
In a statistical analysis, parameter estimates are a crucial part of the
decision-making process. These estimates are a function of the observed data, the
postulated model, and the analysts choice of statistical inference technique. Any
one of these three aspects of a statistical-modeling situation is subject to uncertainty
and doubt: observed data may be incorrectly recorded or subject to rounding
error, a slightly different model may be just as plausible as the postulated model,
and nuisance aspects of modeling such as choosing hyperparameters are subject to
second-guessing. If small changes in these aspects of a statistical modeling problem
create large changes in parameter estimates, the analysis is unreliable. Therefore,
the change in parameter estimates due to perturbation can serve as a barometer for
the robustness of the scientific conclusions.
1


48
3
2
1
*t Hllllllll|l|*|l
5 10 15 20 25 30 35
(a) BasicFITS
5 10 15 20 25 30 35
(b) MaxFITS
(c) BasicFITS (stars connected by
solid line) and MaxFITS (triangles
connected by dashed line)
Figure 2.2: Hills data; c2D(\/ to be quite influential on its fitted value. Plot (b) shows the 35 MaxFITS (i.e. the
Umaxs for directed influence on each of the 35 //(*)) Since no point stands out
as being large, the plot indicates that no fitted value is particularly sensitivety to
variance perturbations. Plot (c) shows both sets of diagnostics. The BasicFITS
values are stars joined by a solid line, while the MaxFITS values are triangles joined
by a dashed line. The closeness of the MaxFITS and BasicFITS values for race 7
indicates that this observation is somewhat self-predicting.


78
Next, we find maximum directed influence on k-j+1 using the results of Beckman
et al. (1987):
v.
oc
A'T
a-*- atk-j+i
ip r ¡3 Xip
0
0
0
oc r p 0 Xy>*-+i
C'max 21| AQTak_>+11|2
= ^¡F (¥>*-+i r'~P (p'k-j+1X')(tpk_j+l r-¡3 X = a24_j+1 ^ 0 rV ^
= ^ II^H2<5*-J+l)
= 2(
n
4-
+
j+i
T/91
Since these expressions are the jth eigensolution of Fl the proof is complete.
This theorem gives added insight into the direction of maximum curvature.
In short, umax is the direction that changes k the most. Similarly, the jth largest
eigenvalue and associated eigenvector specifically correspond to effects on *_+1.
Also, the sizes of the curvatures show that the estimated coefficients for the most
important principal component regressors are also the least sensitive.


Hence, the maximum curvature and its direction are identical to that of directed
influence on fa. Confirmation of this first eigensolution follows:
55
- w) +ft) OM'O
oc (rr1 + ||ri||2/9i H fa (rr[ + nr')) (r far^j
- (2INI2 + IMI2)1 (r *r)
= ||r||2r /?i||r||2r! p¡\\rl\\2Hr1 + Pl\\ri\\2r
~ 2lki||2r ||r||2r + /?i||ri||2r1 +^i||r||2ri
= -/313||r1||2Jffr1+A3||r1||2r1
= 0.
The second eigenvalue is and has multiplicity A: 1. As mentioned earlier,
this implies that there is no unique set of orthogonal eigenvectors to complete the
solution.
3.2.5 Influence on 6
The acceleration matrix for examining both $ and a2 simulatenously has
k + 1 = p non-zero eigenvalues. The form of the eigensolution is unknown. My
numerical studies indicate that there are three unique non-zero eigenvalues, one of
which has multiplicity k 1. The one with higher multiplicity appears to be the
second largest of the three, with a value of
Example
Huynh (1982) and Fung and Kwan (1997) examined data from 20 randomly
sampled schools from the Mid-Atlantic and New England States. The mean verbal
test score (VSCORE) of each schools 6th graders is regressed on staff salaries per
pupil (SALARY), % of white-collar fathers (WCOLR), a socio-economic composite
score (SES), mean teachers verbal score (TSCORE), and mean mothers educational


99
where
dm(u,y(0))
du
o
dy (0U)
d'u
o
s7 =
9m(a;,7((9a;))
dj (Ou)
_ d§u
~ d7
5
0
and |o denotes evaluation at the unperturbed fitted model: u) = u0 and 6w = 6.
Proof of Theorem 4.1 Taking partial derivatives of m we find
dm
da
dm duj
du' da
dm(u, 7(0))
o=o*
07(0)' d' )
+ dm(u, 7(flJ) ^ \
WA' ao'u d'J
V.
Evaluating at the original model completes the proof.
Corollary 4.1
The gradient is given by
rh = Su + J'K's7.
(4.24)
4.4.2 Second-order Assessment
In order to make a general case expression for the acceleration, a large
number of partial derivatives are needed. The following lemma provides an identity
used to derive the algebraic expression for M.
Lemma 4.1
Define the following:
a2 7,(0,,))
a2-,Ae)
p ^
dudu'
0 1 dodo1
0 dudui'
Then,
r = [iT j'] gj + [k iq] p,


116
An interesting consequence of this result is the following: if the observed and
expected information matrices are equal, then the acceleration matrix is proportional
to the acceleration matrix of the likelihood displacement.
4.7 First-order Assessment of the Log Generalized Variance
In this section, some general results for first-order local assessment of the log
of the generalized variance are presented. In addition, the directional derivative for
the variance perturbation of a regression model under ML is derived.
4.7.1 General Form of the Directional Derivative
Theorem 4.5
For the measure
m = log
_ (dhu(0u;y)^_i
80,
the kth element of the vector is given by
d2hu(0;y)
trace
duJkdO'
h
and the jth element of the vector s1 is given by
'82h(0; y)
trace
Additionally, we may write
dOjdO1
h
-l
8U = R(h ) =
Rl i?2
k = l,...q,
j = 1
Rp
vec(h 1),
where each Rj is a q x p matrix of the form
82hjiU](0] y)
dujdG1
j = 1
(4.51)
(4.52)
(4.53)
(4.54)


113
The next step is to construct the building blocks of a second-order assessment.
In order to obtain a formula for K, we present the following result, which appeared
in Cook (1986) as equation (24):
dg{01)
o
L21
where
L =
Ln
£21
Using this result, we find
K =
dl{Ou)
del
*Ho
eei 10
I
0
0
l
,05
O
1
L 22 1*21
0
(4.47)
(4.48)


36
(a) v2 for 9 (b) v$ for 6
Figure 2.1: Hills data; variance perturbation
This is similar to the ith Cooks Distance (Cook 1977):
1 r? h-.
k(l haYs* '
2.5.5 Influence on 6
The eigensystem of F is not closed-form, but it can be found numerically.
As an example, consider the perturbation of the error variances for the Scottish
hills data. There are three non-zero eigenvalues for F: 14.82, 4.91, and 0.37. The
eigenvector associated with the largest eigenvalue was plotted in Figure 1.10(a). It
is similar to umax for directed influence on a2, as given in Figure 1.10(b). The other
two eigenvectors for F are plotted in Figure 2.1.
2.6 Benchmarks for Perturbation Analyses
An analyst using local influence diagnostics will want to know when a
curvature is too big and when an element of vmax is too big. Calibration of local
influence diagnostics was the topic of much debate in the discussion and rejoinder
to Cook (1986), and there is still no consensus opinion on the subject. Much of the
difficulty lies in the fact that the perturbation is introduced into the problem by the


24
where T is the vector for which TT' = (L B22). It can be expressed as
/ \
T = J-
Vc
\
1
-^22 -^21
(2.4)
where c = Ln Li2L22L2\-
2.3 Building Blocks of Local Influence for Regression
In this section, the building blocks of F for a multiple linear regression are
constructed. Subsequent sections apply these results to the response and variance
perturbations used in Chapter 1.
2.3.1 Notation
Unless otherwise noted, the derivations assume the presence of an inter
cept in the model so that the residuals sum to zero. Centering and scaling
the regressors tends to faciliate interpretation, but is not necessary.
The number of regressors is k p 1.
. H = X(X'X)-'X.
Xi represents the ith column of X, and x\ represents the ith row of X.
r = (In H)y represents the residuals.
r represents the residuals from fitting a model to the ith predictor using
the other predictors. These can be considered as values of an adjusted
predictor (i.e. adjusted for relationships with other predictors). Note:
Hrt = and (J i/)r = 0.
X(i) represents the X matrix with the ith column removed. Similarly,
is the regression parameter vector with the ith element removed, and
A() is A without the ith row.


42
reparameterization for which the first pi elements are the same as those of 0:
0 =
1
H
1
1
1
to
1
to
-
This implies that
90
0
d 2 d2
drp[ drp2
(2.24)
and this leads to the following expression for the lower right submatrix of L^:
dO' dO
C'V22
d*l> 2 302
d' dO' dO dcp
-Lq-
dip2 90 d(\j dtp'2
0 M.
U drp2
do'
w'
dO
dcp1
0
02
dip'2
(2.25)
d(p'2 dO' dO d 902902 ^02^02
= 902 902
902^2290'
where all partial derivatives are evaluated at the original model. Meanwhile, the
acceleration matrix for directed influence on tpl is
=
-2a;(¡1 -
0 0
0
)\
2A'9(Ld 1 ^
9K 0 90'
0 0
0
(2.26)
d0\A
90^*'
The invariance of the acceleration matrix for directed influence on (p1 to the choice
of 02 is proved by showing that (2.26) is equivalent to (2.23). Using the inverse of


114
It follows that the acceleration matrix for the profile likelihood displacement
is
I J'K'
I J'K'
I J'K'
I
KJ
+ [; (8) J'] GJ + [s\K Iq] P
0
0
0 d2m(u,7(fl)) |
U (0)' 1
0 0
/-> d2m(u,6) i
U 8086' 1
I
KJ
I
KJ
r -|
0
0
I
I J'K'

0
-2 L
KJ
= -2JKLKJ
= -2A L
= -2A L
Ipi LuL22
0
0
L n L\2
L2i L22
lp 1
-1
0
-^22 -^21 0
*A
-^li ~ Li2L22 L2\ 0
0 0
L A
-2A'(L X-
0 0
0 L22
)A.
This is equivalent to the formula given in Section 2.1, as per Cook (1986).
4.6.5 Kullback-Leibler Divergence
The Kullback-Leibler divergence,
fiv \9))
f(y I K)
m = Eg
?


196
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Linear Regression Models, The Statistician, 46, 71-79.
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Stahel, W. and Weisberg, S., editors, Directions in Robust Statistics and
Diagnostics. Part I, pages 89-100, New York: Springer-Verlag.
Geisser, S. (1993), Predictive Inference. An Introduction, New York: Chapman &
Hall.
Godambe, V. P. (1960), An Optimum Property of Regular Maximum Likelihood
Estimation, The Annals of Mathematical Statistics, 31, 1208-1212.
Godambe, V. P. (1976), Conditional Likelihood and Unconditional Optimum
Estimating Equations, Biometrika, 63, 277-284.
Godambe, V. P. (1991), Orthogonality of Estimating Functions and Nuisance
Parameters, Biometrika, 78, 143-151.
Godambe, V. P. and Heyde, C. C. (1987), Quasi-likelihood and Optimal
Estimation, International Statistical Review, 55, 231-244.
Godambe, V. P. and Thompson, M. E. (1989), An Extension of Quasi-likelihood
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483-491.
Kass, R. E., Tierney, L., and Kadane, J. B. (1989), Approximate Methods for
Assessing Influence and Sensitivity in Bayesian Analysis, Biometrika, 76,
663-674.
Kim, C. (1996a), Local Influence and Replacement Measure, Communications in
Statistics, Part A Theory and Methods, 25, 49-61.


37
analyst. Because the technique does not involve hypothesis testing, critical values
are not appropriate. However, there have been attempts to develop rule-of-thumb
benchmarks, as exist for Cooks Distance and other common influence diagnostics.
In this section, we discuss some general difficulties in developing benchmarks
for perturbation analyses, as well as some issues specific to the local influence
approach. Although it is argued that all sensitivity is relative, some simple
methods for assessing the size of curvatures and the size of the elements of directional
vectors are presented.
2.6.1 The Analysts Dilemma
As mentioned before, the perturbation is introduced by the analyst, and this
implies that he/she must decide whether the effects of perturbation are important,
much the way a researcher decides what effect size is important. Consider a situation
in which the researcher is actually able to give the size or stochastic behavior of
the perturbation. Here, we may concretely measure the effects of perturbation
by whether or not the conclusion of a hypothesis test changes. Alternatively, the
effects can be measured by whether or not the perturbed estimates are outside an
unperturbed confidence ellipsoid. However, despite having a concrete measure of the
perturbationss effects, assessing the practical relevance of the results still falls to
the analyst.
2.6.2 Calibrating Curvatures in Local Influence
In addition to the general limitations of a perturbation analysis, an analyst
using the local influence approach must also contend with the fact that the behavior
and size of the perturbations are undetermined apart from the basic formulation.
Loynes (1986) and Beckman et al. (1987) noted that curvatures do not have
the invariance properties needed for a catch-all method of benchmarking. This
necessitates a reliance upon relative influence.


11
Time
200
150
100
7
11
is


5 10 15 20 25 30
Distance
Time
200
150
100
11
18
A

1000 2000 3000 4000 5000 6000 7000
8001#b
(a)Winning time vs. race
distance
(b)Winning time vs. race
climb
Climb
8000
7000
6000
5000
4000
3000
2000
1000
7
I

11
M*'
5 10 15 20 25 30
Distance
(c)Race climb vs. race dis
tance
Figure 1.5: Hills data; scatter-plots with highlighted cases
1
0.75
x
2 0.5
0.25
0

-*. it. **
§ -0.25
g
-0.5
u
-0.75
-L-lii 1 1 i I > IL_
0 5 10 15 20 25 30 35
Case
Figure 1.6: Hills data; response perturbation; i?max for 6


195
Cook, R. D. (1977), Detection of Influential Observation in Linear Regression,
Technometrics, 19, 15-18.
Cook, R. D. (1979), Influential Observations in Linear Regression, Journal of the
American Statistical Association, 74, 169-174.
Cook, R. D. (1986), Assessment of Local Influence (with discussion), Journal of
the Royal Statistical Society, Series B, 48, 133-169.
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In Stahel, W. and Weisberg, S., editors, Directions in Robust Statistics and
Diagnostics. Part I, pages 47-60, New York: Springer-Verlag.
Cox, D. R. and Reid, N. (1987), Parameter Orthogonality and Approximate
Conditional Inference (with discussion), Journal of the Royal Statistical
Society, Series B, 49, 1-39.
Crowder, M. (1986), On Consistency and Inconsistency of Estimating Equations,
Econometric Theory, 2, 305-330.
Crowder, M. (1987), On Linear and Quadratic Estimating Functions,
Biometrika, 74, 591-597.
Desmond, A. F. (1997a), Optimal Estimating Functions, Quasi-likelihood and
Statistical Modelling (with discussion), Journal of Statistical Planning and
Inference, 60, 77-121.
Desmond, A. F. (1997b), Reply to Comments on Optimal Estimating Functions,
Quasi-likelihood and Statistical Modelling, Journal of Statistical Planning
and Inference, 60, 116-121.
Escobar, L. A. and Meeker, William Q., J. (1992), Assessing Influence in
Regression Analysis With Censored Data, Biometrics, 48, 507-528.
Farebrother, R. W. (1992), Relative Local Influence and the Condition Number,
Communications in Statistics, Part B Simulation and Computation, 21,
707-710.
Firth, D. (1987), On the Efficiency of Quasi-likelihood Estimation, Biometrika,
74, 233-245.
Fuller, W. A. (1987), Measurement Error Models, New York: Wiley.
Fung, W. K. and Kwan, C. W. (1997), A Note on Local Influence Based on Normal
Curvature, Journal of the Royal Statistical Society, Series B, 59, 839-843.
Fung, W. K. and Tang, M. K. (1997), Assessment of Local Influence in
Multivariate Regression Analysis, Communications in Statistics, Part A -
Theory and Methods, 26, 821-837.


85
0.
0.
0.
0.
2 3 4 5
Figure 3.12: School data; X + W pert.; row profiles of |umax| for directed influence
on $ with large elements labeled.
A bubble plot of | Wmax| provides the best graphical display of the information
in umax (Figure 3.13). Each element is represented by a point plotted according to
both the corresponding regressor (z-axis) and case (y-axis). Around each point is a
bubble with radius equal to the absolute value of the element, and so large bubbles
indicate large contributors to the direction of maximum curvature. Again, we see
that the elements corresponding to zi8,2, Zi8)5, £3,2, £3,5, and z2,3 have the largest
influence, as evidenced by having the largest bubbles.


184
Figure 5.11: Corn data; Scheme 3; influence on predicted means- basic curvatures
and maximum curvatures


167
5.9 Application to Corn Crop Data
In this section, the residual variance perturbation and cluster variance
perturbation schemes are applied to an agricultural data set. After briefly reviewing
the data set and goals of the analysis, local influence techniques are applied.
Graphical results indicate that the data set has an influential observation.
5.9.1 Data
Battese et al. (1988) analyzed reported areas of corn crops from 12 Iowa
counties using a variance components model. Keeping the same notation as in
Battese et al. (1988), the model is
Uij A) I- Wj + ftcorn^lij d~ fisoy%2t] d" ^ij j
where
Uij reported hectares of corn in segment j from county i
Xiy = no. of pixels of corn in segment j from county i as per satellite imagery
= no. of pixels of soy in segment j from county i as per satellite imagery
Ui ~ i.i.d. N(0,a2)
Cij ~ i.i.d. iV(0, a2)
i = 1,... 12
j = 1, Hi
{nj = (1,1,1,2,3,3,3,3,4,5,5,6)
J> = 37-
Of particular note in the data set is observation 33. The reported hectares
for this case are exactly the same as those of the previous observation despite
having vastly different covariate values. Battese et al. (1988) felt that the value was
misrecorded, and did not include it in their analysis.


8
is large, then U{ is a key player in producing the largest curvature of LD(w) at the
nu-11 perturbation, i.e. in producing an immediate change in parameter estimates.
The direction of maximum curvature is the main diagnostic of Cooks local
influence methodology. Typically, vmax is standardized to have length one and is
then plotted against indices 1,2,..., q. If any elements are particularly large relative
to the others, they stand out in the plot. The analyst then may want to take steps
to alleviate the sensitivity to the perturbation. What, if any, steps to be taken
depend upon the perturbation scheme and the particulars of the data analysis.
For some models and perturbation schemes, the formula for vmax is known
explicitly, while in other situations it requires some moderate computations. In
analyses such as regression, vmSLX is typically a function of familiar quantities.
1.4 Perturbations in Regression
Consider the normal linear regression model:
Y = X/3 + e, (1.10)
where Y is a n x 1 vector of responses, X is a known nxk model matrix of rank k,
/3 is a k x 1 vector of unknown parameters, and e is a n x 1 vector of independent
identically distributed normal random variables with mean 0 and variance c2 > 0.
Hence, Y ~ Af(X/3, a2In), and 0 = (ft,..., ft, a2)'.
1.4.1 Scheme 1: Measurement Error in the Response
In order to examine the effects of measurement error in the response,
Schwarzmann (1991) introduced perturbations to the observed data. He considered
fitting a model to the observed data set with slight perturbations: ¡ji + Ui, for
i = 1,..., q. Hence, w0 is a vector of zeroes for this perturbation scheme. By letting
q n, all of the observations can be perturbed simultaneously. This allows the


Next, we show that directed influence on (¡>x = x'O is not a function of fc2,..., kp.
From equation (2.23), we find
45
f"] = 2^A'e'1/2K)
l
|fci||2J-||fci|
V
0 0'
0 I
(p^1/2a}
2 :A'aL01/2K
|fci||2 *
1 0'
0 0
K'L91/2Ae
2 A'gL^k^L^Ae
lilil2
2 A'aLalxx'LelAe.
||fc-1'2 9 6
(2.28)
The single eigensolution of this rank-one matrix yields the final solution for Cmax
and X)max Cl
The construction of 0 is worth some note. The premultiplication of 9 by
K'Lg orthogonalizes the parameters, in the sense that the off-diagonal elements
of the observed information matrix for 0 are zero. (Note that the orthogonalization
is with respect to the observed information matrix as in Tawn (1987), rather
than the expected information as in Cox and Reid (1987).) First, a = Le 6
is a parameterization with an observed information matrix equal to the identity
matrix. Then, K is constructed to make orthogonal contrasts of a, with the first
column corresponding to the linear combination of interest. Finally, by scaling the
columns of K to all have the same length as kx, K becomes a scalar multiple of
an orthonormal matrix, and this leads to the algebraic simplifications. Also, if
x'0 = 6j the local influence diagnostics provided by the theorem are the same as
those provided by (2.3), as per Beckman et al. (1987).


127
is related to a linear predictor r/ = x'/3 through a link function, g:
M = V-
4.8.2.2 Perturbation Schemes
One perturbation is to introduce case-weights, or equivalently, variance
perturbations. A set of n perturbations can be introduced into the estimating
functions in a multiplicative manner:
K{0- y) = ~D{u})V{n)~l{y n(0))/o2.
Writing the jth perturbed estimating function in summation notation shows how the
perturbations weight the contributions of the individual observations:
The matrix A is then given by
dK{0\y) 1 dn' y-n
M o o*d0 V(/i)V
where D(^^r) is a diagonal matrix with ith diagonal element /x*)/V(|i).
If the response is continuous, then a perturbation can be made to assess
additive errors in measurement. The perturbed estimating functions are then
M0;v) = 1(y + w n{0))/a2,
which leads to
1 dhu{0;y)
a2 dui'
dy,'
~dG
VW
o
Other perturbations can also be introduced, including errors in the explanatory
variables and perturbations to the linear predictor.


also requires a second-order assessment. The measure does not depend on u;
explictly, so 8. vanishes. Also, 7(,) = 0U, implying
115
Ssy
<9ra(u>,7(0w))
O~f(0.)
= ~^TE0
00.
dE§
f(v\o.)\
L(Ou\y)
00,.,
= -E
OL(0.\y)
00,
= -Ei [0]
= 0.
Proceeding with a second-order assessment, we first note that K
also make the following observation:
92 E
00.00, 9
f(y \ouy
o2
00.00,
teb [^(0>;y)
= -E6
O2L(0.-y)
00.00,
Evaluating this at the null perturbation implies
PMv)
~n~ { d9ff
and the acceleration matrix follows:
J 0
F = I J'K'
I -A L
= A L 1 (E*
I
KJ
1
0
0
I
.. _i
-*
0
1
CO
-L A
O2L(0; y)
0000'
L *A.
Ip- We
(4.49)
(4.50)


172
Figure 5.3 shows the directional vectors vmax for directed influence on the
individual BLUPs. Typically, it would be unnecessary to examine all of these plots,
especially if there are a large number of random effects in the model. Nonetheless,
all plots have been included here to more fully demonstrate their behavior. Upon
inspection, maximum perturbation of is generally achieved by perturbing cases in
cluster i. This is not surprising given the closed-form results for the one-way random
effects model in Section 5.8.3. Only in plots (a) and (g) do we see an observation
in another county exerting influence. In both cases, observation 33 is the influential
out-of-cluster data point.
Figure 5.4 contains local influence diagnostics associated with point inferences
for linear combinations of fixed and random effects. Recall that in Chapter 2,
MaxFITS was proposed as a measure of how sensitive fitted values are to residual
variance perturbation, while BasicFITS was proposed as a measure of how much a
single observation determines its fitted value. These diagnostics are Cmax and C\,
values for 6\ respectively. Figure 5.4(a) shows MaxFITS (stars) and BasicFITS
(triangles). The values for MaxFITS indicate that the prediction of cases 33
and 34 are noticeably more sensitive to pertubation than the other predictions.
The sensitivity of prediction 33 is largely a function of perturbing this case, as
demonstrated by the fact that its BasicFITS value is nearly as large as its MaxFITS
value. This is confirmed by t>max for directed influence on the fitted value of case 33
- element 33 is by far the largest contributor (plot (c)). However, the sensitivity of
case 34s prediction is not largely a functon of perturbing case 34, as demonstrated
by a moderately sized BasicFITS value. The source of the influence is in fact case 33,
as is evident in the plot of umax for directed influence on the fitted value of case 34,
(plot (d)). In short, the fitted values for observations 33 and 34 are both sensitive
because they are greatly influenced by the models assumption that observation 33
is just as reliable as the rest of the data.


83
(a) l^maxl for CT2
(b) l^maxl for £ (c) |2| for
Figure 3.9: Hills data; Model II; X + W pert.; column profiles for directional vectors.
DISTANCE values are connected by solid line. CLIMB-RATE values are connected
by a dashed line.


39
To further develop this idea, let us define the size of a perturbation to be
its Euclidean distance from the null perturbation (i.e., ||w u>0||). Suppose further
that we have perturbations in two different directions: in direction and ug
in direction ug. If \C\ is k times larger than |Cg|, then perturbation u>g must be
\/k times larger than u>a to have the same effect on the likelihood displacement.
Equivalently, in order to perturb the estimates to the edge of the confidence ellipsoid,
it would be necessary to travel \/k times farther in direction vq in SI as in direction
v. This provides a simple interpretation for the relative size of two curvatures.
2.6.2.3 Internal benchmarks for curvatures
Building on the interpretation given above, curvatures can be benchmarked
based on their relative size to some sort of average curvature for a given scheme
and fitted model. For instance, the average of the q basic curvatures for the fitted
model can serve as a benchmark. Alternatively, the average curvature could be
computed by (1) allowing the squared elements of v to have a Dirichlet(o:i.. .aq)
distribution with all equal to one, and then (2) finding the expected curvature.
This would be equivalent to finding the average approximate displacement at a
radius of \/2 from u>o (Cook 1986). If such a benchmark is difficult to compute, a
Monte Carlo estimate can be calculated. Because these benchmarks are computed
using curvatures from the fitted model, they result in comparisons that are internal
to the fitted model. The conformal normal curvature of Poon and Poon (1999)
would also be an internal way to assess the sizes of curvatures.
2.6.2.4 External benchmarks for curvatures
It is entirely possible that all of the curvatures of a fitted model are large
due to the nature of the data set, and using internal benchmarks will not alert
the researcher to such a phenomenon. To address this, a benchmark that is not
dependent on the observed data is needed. To this end, we might utilize the
expected value of the curvature in direction v, using 0 in place of the unknown


12
0.25
-0.5
0.75
-1
(a) t7max for directed influ
ence on /?dist
(b) t>rnax for directed influ
ence On /?climb
Figure 1.7: Hills data; response perturbation
1.4.1.2 Influence on a subset of parameter estimates
Plotting Umax for the profile likelihood displacement, LD^6l\cj), can indicate
which components of uj affect a subset of parameters. Examining influence on
9\ (f2 results in the same vmax as the original likelihood displacement, i.e. it is
proportional to the residuals. The directions of maximum curvatures for = /3dist
and §i = /3ciimb are shown in Figures 1.7(a) and 1.7(b). The first plot indicates that
small measurement error in race lls winning time is most influential on /5 plot of vmax for /3ciimb> elements 7 and 11 stand out. The fact that the two elements
have opposite signs can be used for interpretation. They indicate that maximum
curvature is primarily achieved by perturbing winning times 7 and 11 in opposite
directions (i.e. making one observation smaller and the other larger). Thus, if the
record winning times for races 7 and 11 were slightly closer together or farther apart,
this would affect the slope estimate for CLIMB.
1.4.1.3 Plotting options
Note that the standardized vmax is only unique up to sign change. For
example, Figure 1.8 displays the same standardized vector as Figure 1.6, but with
all of the elements signs switched. This suggests an alternative plot in which the


186
Because hi)UI(6i,y) does not depend on 02,u>, there are some simplifications
in the computational formulas. For instance, using results on partitioned matrices
(Searle 1982),
h 1 =
-i -i
/in 0
/l21 /l22
-1
h
11
Partitioning A leads to a simplification in J:
d0
du'
-h
Ai
A2
0
-/t22 tl2\h\x h>22
-Ki Ai
h22 (h2ihn Ai A2j
(6.1)
(6.2)
6.2 Synopsis and Additional Future Research
The local influence approach to diagnostics is largely unknown, owing in part
to the lack of introductory resources on the technique. It is hoped that the extended
and detailed discussion in the first two chapters of this thesis have shed some light
on this powerful and practical methodology. The diagnostics derived for perturbing
the entire X matrix in Chapter 3 give previously unavailable detail about how
collinearity and overfitting can affect estimation. The results should encourage
researchers to utilize perturbations other than just case-weights and place more
emphasis on finding closed-forms for the resulting diagnostics. Because analysts are
unaccustomed to using directional vectors, having interpretable formulas for special
cases will help them understand and appreciate the results.


117
Similarly,
s7 = h(h ) =
h\ /12
K
vec(h 1),
(4.55)
where each hi is a p x p matrix of the form
v d2hiiW(0;y)
i dOdO'
Note that the matrices Ri and hi appeared earlier in the computational
formula for P (Theorem 4.4).
Proof of Theorem 4.5
To begin, we present the following two matrix differentiation results for a matrix A
and a scalar t (Searle et al. 1992):
dA
-1
dt
Combining these two identities yields
r 4
| = trace
A-1
dt
d .
log I A I = trace
trace
= trace
(-aha-'^a-1)
dA ,
dt
Using (4.58) we find
d
duik
m{oj, OJ) = trace
d2hu(0;y)
duJkdO'
9=6u
. dhu(Qu] y). 1
1 W, J
(4.56)
(4.57)
(4.58)
and evaluating at u> = u;0 and 0 = 0 yields (4.52). Expression (4.53) follows from a
similar argument.


149
The implication is that the gradient vector vanishes. Proceeding with a second-order
assessment, we note
S =
d2m(ui,j(6)) i d2m(0J,7(fl)) I
dudio' 1 duid'tiey 1
d2m(ui,7(g)) | a2m(o),7(g)) i
di(6)du' 1 d7(8)d-y(8)' 1
K =
56,
= J.
0 0
0 s
77
Using (4.32), the acceleration matrix is then
M =
Iq J'K'
KJ
= J'S^J
= A'h1 S^h1 A
= -2A'k-'a2L^
8080'
h *A
(5.26)
= 2A'h A.
where h 8h(u, /3; y)/80'\o- The matrix h is derived as follows. Taking derivatives
of the estimating functions with respect to 0 we find:
X'R~l(y X/3 Zu)
Z'R~\y X/3 Zu) G-1u
X'RlX X'RlZ
Z'RlX Z'R'Z + G 1
8h(u, /3; y) 8
80' ~ 80'
(5.27)
The expression is not a function 0, so (5.27) is in fact h. It can be shown by
inverting this matrix that the variance matrix of the BLUEs and BLUPs is (Searle


92
where Ew(/j) = yiuj. None of these three measures depend on w directly. The first is
a linear combination of the perturbed parameters. Here, = x'0w is a scalar.
The second measure is the ratio of the ith unperturbed parameter estimate to the ith
perturbed estimate. Again, 7{0W) = 0iu is a scalar. The third measure resembles a
Pearson goodness-of-fit statistic based on the perturbed estimates. Since it is not yet
specified how the means, /t^, depend on the estimates, we simply write 7{0J) 0U.
The following measures are more application-specific, in that they depend on
how inference is performed and perhaps even upon the perturbation itself:
ra4 = LD(u) = 2[L(0; y) T(0W; I/)]
m5 = LDW(u) = 2[L(0lt02\y) ~ L(0llv, g(0lul); y)]
m6 = LD*{w) = 2[LU(0U; y) Lu(0; y)]
m7 = E§
f(y 1 A) \
m8 = log
_ (dhujOuiy)^
d'
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
The first of these measures is the likelihood displacement, for which 7(0W) = w.
The second is the profile likelihood displacement, which has 7(0W)' = (l0, g(0il)'),
where g{0uo) maximizes the profile likelihood of 0iiO with respect to 02- The third
measure is a likelihood displacement with a moving frame because the base
likelihood of the measure is the perturbed one. Obviously, these three measures are
intended for use when estimation is performed by maximum likelihood.
The influence measure m7 is a Kullback-Leibler divergence between the
densities f(y | ) and f(y | 0W). The quantity is non-negative, achieving its
minimum value of zero when the two densities are equivalent. Here, 7(0W) = ^.
The last measure addresses sensitivity of asymptotic standard errors. Recall
that the determinant of a matrix equals the product of its eigenvalues, and when
the matrix is a variance-covariance matrix for a vector of estimates 0, the squared


14
-i* -i-
(a) All elements numbered (b) Largest elements numbered
Figure 1.9: Hills data; |vmax| as a star plot
1.4.2 Scheme 2: Perturbations to Measurement Precision
Another assumption that can be perturbed is that of equal variance (Cook
2
1986). This is accomplished by letting V(yi) = for i = 1,...,n. Here the null
perturbation is u?0 = ln-
With this perturbation scheme, we deem that some observations are observed
with more precision than others. The resulting perturbed MLEs weight each case
according to its perturbed variance. If certain cases are influential, then even a
slight down-weighting or up-weighting of them produces large displacements.
This perturbation scheme has an important connection to the widely-used
influence measure Cooks Distance (Cook 1977). Recall that the ith Cooks Distance
is a measure of the change in regression parameter estimates that results from
deleting the ith observation. Clearly a deleted observation provides no information for
finding ¡3. Similarly, as the ith perturbation is decreased from 1, the ith observation
provides less and less information to the estimation process owing to its increased


162
The formula has several implications. First, the acceleration matrix for the
influence measure PLD(w) can be written as
M = 2A'h lA

hn
h12
X'RlD{ei)
D(ei)'R-lX 0
21
22
L J
h
h
0
= 2D{ei)' RTl X (h11)X'R~1D(ei)
= 2D(ei),R-1X{X'V-1X)-1X'R-1D(ei).
This matrix has at most pi distinct non-zero eigenvalues regardless of the number of
random effects. In addition, using the results given in Section 5.5.3, the acceleration
matrix for influence measure PLD^\(jj) is equivalent to (5.45). Finally, the
acceleration matrix for PLD^(uj) is
r -i
, -1
hn 0
)
X'RlD{ei)
M =
D(ei)'R~1X 0
(h -
0 0
0
= 2D(ei)'R-1X{(XlV-1X)-1 (X1 R~lX)~x)X'R~lD(ei).
(5.46)


91
inference method would be to allow the value of the hyper-parameters in a Bayesian
analysis to vary. Though some might argue that the hyper-parameters are part of
the model, their values are often chosen for computational convenience.
For the purposes of this chapter, we assume that the perturbation scheme
results in a set of p perturbed estimating equations,
M0;y) = o, (4-5)
which have solutions 9w. It is also assumed that there exists a null perturbation, w0,
whose estimating equations are identical to the original ones. Finally, it is assumed
that there are no constraints on w that would limit their values in a neighborhood of
w0- For example, constraining the space of variance perturbations to positive values
is allowed. However, constraining the perturbations to sum to one is not allowed.
These kinds of constraints could be accommodated, but are not considered here.
4.1.3 Influence Measures
The final item to specify is a quantity whose sensitivity to perturbation is of
interest. This influence measure serves as a barometer for how perturbation effects
inference. The measure can depend on w directly and indirectly via some r x 1
vector of functions of the perturbed parameter estimates, 7(9W). Some influence
measures, m(u>, 7(0^)), may be appropriate regardless of the method of estimation
and the perturbation, while others may be more application-specific.
Examples of rather general objective measures are the following:
mi = x'Ou
(4.6)
9i
rn2 =
Oi.
(4.7)
/ J \T ( \
t V(^)
(4.8)


CHAPTER 3
PERTURBATIONS TO THE X MATRIX IN REGRESSION
In this chapter, the influence that infinitesimal errors in the X matrix can
have on parameter estimates in a linear regression is addressed. First, previous
results for single column and single row perturbations are reviewed. This is followed
by a generalization of these results to perturbing all elements in the X matrix. It is
shown that the eigensolution of F is related to principal components regression,
collinearity, and outliers. Finally, new graphical techniques are presented with the
results applied to the Scottish hills data set and an educational outcomes data set.
3.1 Perturbations to Explanatory Variables
Letting k denote the number of predictors, Cook (1986) proposed that an
nA;-sized perturbation can be incorporated into the design matrix in an additive
fashion. These perturbations represent fixed errors in the regressors and should
not be confused with random errors-in-variables techniques (Fuller 1987). The
perturbation scheme can be used to assess the effects of small rounding or truncation
errors in the explanatory variables. Upon perturbation, the X matrix becomes
k^n+1
^2n+l
^knn+1
U>2
^n+2
^knn+2
U3
...
^knn+3
Un
U2n
^3 n
Ukn
The use of a single subscript implies u> = vec(LU), where the vec operator stacks the
columns of the matrix into a single column. When particular elements of W greatly
affect parameter estimation, it indicates that the corresponding element of X is
49


101
Theorem 4.2
The acceleration of inuence measure m(u>, 'y(0UJ)) in direction v at the null pertur
bation is given by
d2m(u, 7(0J)
A =
= v
da2
,d2m(u>, 7(0W))
dud u'
v
= v'Mv
(4.26)
T
\
v'
Iq J'K'
S
Q
+ [; 0 J'] G J + [s\K 0 Jj P
KJ
/
v,
where
s
s
32m(w,7(0)) I
d2m(ui,j(6)) i
*-*ULJ
duidui 'u
duid'Tf(9)1
S'
s
52m(w,7(0)) i
52m(o;,7(0)) i
d-y(9)d'y (ey 1
^ uj'y
-77
(4.27)
Alternatively, we may write
A = v'
= v
[ U J K
S
Iq
lL J
/
KJ
Iq J'K'
S
h
V
KJ
\
+
i=i
/
(4.28)
+ J'[J2s,iGi]J + 1£^Ki)Pi
1=1
1=1
/
where s7i is the ith element of s7 and is the ith column of K.
Proof of Theorem 4.2 First we define the following (q + r) x 1 vector:


112
likelihood has not been specified, we simply write 7(0^,) = 0W. It follows that
= 0.
8171(00,7(0^))
o 91(0^; y)
ydL(0] y)
dl(0uj)
3
Q5
CO
0
0 90
Hence, the gradient vector is zero. Proceeding with a second order assessment, we
first note that K = Ip. It follows that
M =
Iq J'K'
I J'K'
KJ
I
KJ
+ [; (8) J'] GJ + [s\K (8) Iq] P
I -A L
-1
-1
I -A L
= A L \-2L)L *A
= 2A'_1 A.
1
0
0
I
0
d2m(ui,6) 1
deas1 1
3
-r1
1
0
0
i
0
-2 L
1
t-i:
1
E>~
1
This is equivalent to the result given in Section 2.1, as per Cook (1986).
4.6.4 Profile Likelihood Displacement
The profile likelihood displacement,
7( also requires a second-order assessment. First, the influence measure does not
depend on 00 explictly, so sw vanishes. Also,
s7 =
_ dm(u,7(0u))
d~r (0u)
= -2
dL(j(0u))] y)
d~f{0(ij)
= -2
dWy)
80
= 0.
This implies that the gradient vector is zero.


xeuiA
133


.

. V.

* - 9 + \r -W-
1 *

-
-B -6 -4-20 2
h
Figure 4.8: Leaf blotch data; Model II; vmax vs. linear predictor


179
The small BasicFITS values indicate that none of the counties is self-predicting: not
even county 12.
5.9.5 Scheme 3: Alternate Cluster-based Variance Perturbation
The alternate cluster variance scheme can also be applied to the data set. The
acceleration matrix of the likelihood displacement has twelve non-zero eigenvalues.
The first two eigenvalues appear in Figure 5.9 as index plots and star plots, and both
vectors implicate elements 5 and 11 as being influential. The other ten eigenvectors
are dominated by a single unique element, and are not shown.
Figure 5.10 shows the directions of maximum curvature for directed influence
on each of the individual BLUEs. Although a couple of elements exceed the 2/\/l2
benchmark, the plots are relatively noisy. The small value for element twelve in all
of the plots suggests that perturbing county 12s random effect variance has little
effect on the fixed effects estimates.
Finally, Figure 5.11 provides the BasicFITS/MaxFITS plot for the alternate
cluster perturbation. The first thing we notice about the plot is that the BasicFITS
values are nearly as large as the MaxFITs values. This suggests that each fa is most
sensitive to the perturbation of V(ui). We also recall that for the simple random
sample, the ith element of umax for fa was roughly g 1 times larger than the other
elements. Hence, Figure 5.11 suggests that this dominance may be maintained
in more complicated models. If this is true, it may be sufficient to only plot the
MaxFITS values. For the corn data, the MaxFITS values indicate that the predicted
means for counties with large BLUPs (counties 5 and 11) are most sensitive to the
perturbation of the cluster variance.
5.9.6 Conclusions and Action
Having examined these plots, the analyst should be very wary of observation
33. Scheme I indicates that if its precision varies from that of the other segments


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43
(2.25),
de
0 0
de'
de d
0 0
dxi>'
1
. w
rH js,
1 ^
O
di¡)
d' dx¡y
i
o
-S- 1
s -

de
d de
dfa
de
d'
o
dtj>2 d2
dtp[ dtp '2
o
o
d_fad&_
di¡) d(f>
0 0
Ipi
d'2
dtPt
0
d2
dtp'2
d2 T -1 dtp2
U dtP'2 ^>22 dl¡)'2
dff_
d O O
0 L~
de
d(f)
de*_
d(¡>
(2.27)
Substituting (2.27) into (2.26) completes the result.
2.7.1 Directed Influence on a Linear Combination of Parameter Estimates
As mentioned in Escobar and Meeker (1992), a specific application of these
results is to consider directed influence on a linear combination of parameter
estimates, x'e. There are many ways to reparameterize the model such that x'O is
one of the new parameters. For example, assuming that the first element of x is
non-zero, can be defined as A0, where
^i *E(i)
A =
Opi Ipi
where x\ and X(i) are the first and remaining elements of x, respectively (Escobar
and Meeker 1992). Using equation (2.23), the acceleration matrix for directed
influence on i = x'e can be computed, and the resulting diagnostics are invariant
to the choice of fa,fa. By using a prudent choice of new parameters, we can
obtain easy-to-compute formulas for the single eigensolution of 6\ as per the
following theorem.


164
5.8.4 An Alternative Cluster Variance Perturbation Scheme
In this section, the variance perturbation scheme proposed by Beckman et al.
(1987) is addressed. For the one-way random effects model, the perturbation is
V(ui) = As each Ui approaches infinity, V(u) > oo and V(?/i) > oo. Clusters
perturbed to have higher random effects variance are deemed to provide less reliable
information for inference. Also, the correlation between observations in the same
cluster approaches 1 as > oo:
ac + a2/ui
Thus, observations from clusters that are perturbed to have higher random effects
variance are deemed more alike, while observations from clusters perturbed to have
smaller random effects variance are deemed less alike. This is in contrast to the
perturbation given previously, which preserves the intra-cluster correlations. Also,
the alternate perturbation does not reweight the clusters entire contributions to the
likelihood. Since it does not correspond to a cluster weight, it provides diagnostic
information about the assumption of equal cluster variance rather than the influence
of clusters.
By using V(uj) = rather than V(u) = u;(r;?, we can obtain umax for
this perturbation from results for the first cluster perturbation. Using the partial
derivatives from Section 5.8.3, we may derive
A =
0
-G_1D(u)


100
for
Ti
G!
Pi
^rqxq
r2
Grpxp
g2
Ppqxq
P2
r r
Gr
. pv.
and where J and K were defined in Theorem 4.1.
Proof of Lemma 4.1 Consider:
d'yi(0u) d6u
<927(flg;)) d
dujdu)1 duj
d_
du>
d0
du)
d' dtj3'
djijOu,)
d'
du ^&rt(u)
du)
7 + E
j=i dOju
d2eju
dujduj'
dud[J d'
+
^(81,
dO,,
Evaluating this equality at the null perturbation yields
Ti = J'GiJ +
dyt(u)
Iq
du
0
Stacking the to form T completes the proof:
826
lOL
duiduj'
828,
pui
duidu'
J'GiJ
071 (M
881
Iq
0
P
r =
+
J'GrJ
07r()
881
O
l I
P
= [Ir J'] GJ +
&n(0*) t dji(0u) r
* n TS * n
86
892u
812(8qj) r &Y2(0u) t
88lu 862^ lq
J 0
(4.25)
= [/, J'] GJ + \K 0 /,] P


50
influential. This allows the analyst to identify specific combinations of explanatory
variable values that influence estimation. As will be shown in Section 3.4, sensitivity
of the regression parameter estimates to these kinds of perturbations is greater when
there is collinearity amongst the regressors and when the model is calibrated to fit
the observed data too closely.
3.2 Perturbations to a Single Column
The special case of only perturbing a single column of X was discussed
in Cook (1986), Thomas and Cook (1989), and Cook and Weisberg (1991). This
section recreates their results in detail.
3.2.1 Building Blocks
Here and in the following sections, represents a vector with a 1 in the ith
position and zeros elsewhere. The length of the vector depends on the context of its
use. Note that for of size k,
HX'Xr'di = y^jp (3-1)
and
X&'Xr'di = jjA¡jr (3.2)
where r contains the residuals from regressing X{ on the other columns of X.
Without loss of generality, suppose perturbations are added to the n values
of the first predictor (i.e., Xx becomes X\ + a;). The perturbed likelihood is
Lu,(0; y) oc log ^j(y -X0- pxu)'(y -Xp- ftw)
= log a2 (y'y 2 Piu'y + ftVw).
(3.3)


181
Figure 5.8: Corn data; Scheme 2; local influence on predicted means-cluster-based
BasicFITS and MaxFITS
(c) v2
(d) |t>2| star-plot
Figure 5.9: Corn data; Scheme 3; local influence on 6


192
Solution for Random Effects
Effect
CNTYNUM Estimate
SE Pred
DF
t
Pr > |t|
CNTYNUM
1
2.18380161
7.35943306
23
0.30
0.7693
CNTYNUM
2
1.47455727
7.30784520
23
0.20
0.8419
CNTYNUM
3
-4.72910256
7.26641831
23
-0.65
0.5216
CNTYNUM
4
-2.76387625
6.88938048
23
-0.40
0.6920
CNTYNUM
5
8.36866398
6.42653277
23
1.30
0.2057
CNTYNUM
6
4.27362586
6.52255117
23
0.66
0.5188
CNTYNUM
7
-2.70476376
6.50319394
23
-0.42
0.6813
CNTYNUM
8
1.15642520
6.47806611
23
0.18
0.8599
CNTYNUM
9
5.02562759
6.21203551
23
0.81
0.4268
CNTYNUM
10
-2.88274430
5.92509787
23
-0.49
0.6312
CNTYNUM
11
-8.65058657
5.88291972
23
-1.47
0.1550
CNTYNUM
12
-0.75162907
5.69287011
23
-0.13
0.8961
Tests of
Fixed
Effects
Source
NDF
DDF
Type III F
Pr > F
PIX.CORN
1
23
31.80
0.0001
PIX.SOY
1
23
0.20
0.6574


APPENDIX B
MODEL FITTING INFORMATION FOR CORN CROP DATA
The MIXED Procedure
Class Level Information
Class Levels Values
CNTYNUM 12 1 2 3 4 5 6 7 8 9 10 11 12
REML Estimation Iteration History
Iteration Evaluations Objective Criterion
0 1 243.72248590
1 2 242.59476305 0.00000000
Convergence criteria met.
Covariance Parameter Estimates (REML)
Cov Parm
Estimate Std Error
Z Pr > |Z|
CNTYNUM 63.28953258 75.07490859
Residual 297.72802207 84.54939011
0.84 0.3992
3.52 0.0004
Asymptotic
Covariance Matrix
of Estimates
Cov Parm
Row
C0VP1
C0VP2
CNTYNUM
1
5636.2419002
-2123.731245
Residual
2
-2123.731245
7148.5993683
Model Fitting Information for HECTC0RN
Description
Value
Observations 37
Res Log Likelihood -152.541
Akaikes Information Criterion -154.541
Schwarzs Bayesian Criterion -156.068
-2 Res Log Likelihood 305.0826
Solution for Fixed Effects
Effect
Estimate
Std Error
DF
t
Pr > |t|
INTERCEPT
120.74028537
3.78206463
11
31.92
0.0001
PIX.C0RN
25.76987997
4.56957580
23
5.64
0.0001
PIX.S0Y
-2.04732793
4.55652013
23
-0.45
0.6574
191


175
(a) BasicFITS and MaxFITS (b) Cmax and Cb for predicted
mean hectares per segment
(c) Umax for y33 (d) Umax for J/34
Figure 5.4: Corn data; Scheme 1; directed influence on predictions


CHAPTER 5
LOCAL ASSESSMENT OF PREDICTIONS
In this chapter, local assessment in prediction problems is considered. First,
prediction is motivated by two methods: first principles and predictive likelihood.
This is followed by a brief discussion of influence measures for estimation and
prediction. Then, predictive inference and existing local influence techniques for
linear mixed models are reviewed. These techniques are then extended to assess the
effects of perturbations on predictions, and three perturbations schemes are applied
to an example data set.
5.1 Prediction via First Principles
Consider the problem of predicting a vector of random variables, u.
Among predictors u*, = E(it) minimizes the mean squared-error of prediction,
MSEP(u*) = E(m u*)2. The standard error associated with this prediction
is v/MSEP( u) = y/V{u). However, prediction is typically not so simple; it is
usually performed using information from a sample and in the presence of unknown
parameters.
5.1.1 Prediction After Data are Observed
Suppose now that prediction of u takes place after observing data y. If
the two random vectors are correlated, then an improved inference can be made
about u. Here, the conditional mean, E(w|y), can serve as a point prediction.
This prediction minimizes the conditional mean square error of prediction,
CMSEP(u*) = E(w u*|y)2. Since it minimizes the CMSEP for given y, it also
minimizes the (marginal) MSEP. This best predictor, w, differs from in that it is a
random variable, rather than a constant. Also, the best predictors MSEP can be no
134


25
2.3.2 Acceleration Matrix
The ML estimates of (3 are the same as those of ordinary least squares,
¡3 = (X'X)~1X'y and the ML estimate of a1 is The p x p estimated
asymptotic variance-covariance matrix of the MLEs is
a2 (x'xy1 0
Partitioning A' into [A^ A'ff2], the acceleration matrix is given by
F = 2A'(_1)A
i 4cj4
= To2A^ {XX'Y A/3 + A;2 A.2.
Tb
2.3.3 Acceleration Matrix for Directed Influence on a2
The acceleration matrix for directed influence on a2 will utilize the matrix
L 1 B22, which is given by
0 0
O'
n
This implies that the acceleration matrix for directed influence on a2 is
= 2A'( B22)A
4d4 ,
= a;2aCT2.
(2.5)
n
Noting that A^.2 is in fact a?xl vector, there will be a single eigenvalue and
eigenvector given by
Ai
Ui oc A'^2
These quantities are the maximum curvature and its direction, respectively.


17
Figure 1.11: Standardized vector of DFBETAS for /3ciimb
(1.11)
Here, ha is the ith diagonal element of the hat matrix X{X'X)~lX\ is the ith
residual, and is the ith residual from regressing Xj on the other columns of X.
It can be shown (see Section 2.5) that t>max for directing the effects of the variance
perturbation on ¡3j is proportional to
(1.12)
The elements of the two vectors differ only by a factor of 1 fact, standardizing
the vector of 35 DFBETAS for /^umb to length 1 and plotting them (Figure 1.11)
produces a figure nearly identical to Figure 1.10(c).


4
y
Figure 1.1: Schematic of a plane curve with its tangent line. The angle between the
tangent line and the z-axis is <¡>.
1.3.1 Curvature of a Plane Curve
Curvature is a measure of how quickly a curve turns or twists at a particular
point (Swokowski 1988). Let (f(t),g(t)) be a smooth plane curve. The arc length
parameter s of such a curve is the distance travelled along the curve from a fixed
point A to another point on the curve (Figure 1.1). Letting denote the value of t
that yields point A, s can be computed as
s(t)= A/'(ii))2 + (p'(tt))2)5d, (1.3)
JtA
and we may express the curve in terms of s: (/(s_1(s)), <7(s-1(s))). Additionally, let
T(s) be the tangent line at a point on the curve, and let 0 denote the angle between
T(s) and the x-axis.
Curvature is defined as the absolute value of the rate of change in 0 with
respect to s:
(1.4)
A plane curve that is turning quickly at (f(t),g(t)) has a larger curvature than a
curve that is nearly straight at that point.


19
2.5
1.5
0.5
5 10 15 20 25 30 35
5 10 15 20 25 30 35
(a) Basic curvatures
(b) Cooks Distances
Figure 1.12: Hills data; CVs for variance perturbation and Cooks Distances
Kim 1998; Rancel and Sierra 1999). A number of articles have also appeared on
applications in generalized linear models (GLMs) (Cook 1986; Thomas and Cook
1989; Thomas and Cook 1990; Thomas 1990; Lee and Zhao 1997) and survival
analysis (Pettitt and Daud 1989; Weissfeld and Schneider 1990; Weissfeld 1990;
Escobar and Meeker 1992; Barlow 1997; Brown 1997). Other applications include
linear mixed models (Beckman et al. 1987; Lesaffre and Verbeke 1998; Lesaffre et al.
1999), principal components analysis (Shi 1997), factor analysis (Jung et al. 1997),
discriminant analysis (Wang et al. 1996), optimal experimental design (Kim 1991),
structural equations (Cadigan 1995; Wang and Lee 1996), growth curve models
(Pan et al. 1996; Pan et al. 1997), spline smoothing (Thomas 1991), Box-Cox
transformations (Lawrance 1988; Kim 1996b), nonlinear models (St. Laurent and
Cook 1993; Wu and Wan 1994), measurement error models (Zhao and Lee 1995; Lee
and Zhao 1996), and elliptical linear regression models (Galea et al. 1997).
Many authors have presented modifications of, extentions to, and theoretical
justifications for the basic methodology (Vos 1991; Schall and Gonin 1991; Schall
and Dunne 1992; Farebrother 1992; Wu and Luo 1993a; Billor and Loynes 1993;
Kim 1996a; Fung and Kwan 1997; Lu et al. 1997; Cadigan and Farrell 1999; Poon


Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
190
SALARY
VCOLR
SES
TSCORE
MOMED
Dfbetas
Dfbetas
Dfbetas
Dfbetas
Dfbetas
0.1527
-0.0882
0.0679
-0.0276
0.0339
-0.0660
-0.1443
0.0717
0.0125
0.1444
0.0734
-0.0199
0.0554
-0.2036
-0.2487
-0.0033
0.0428
-0.0220
-0.0051
-0.0563
0.0769
-0.1122
0.1070
-0.0453
0.0394
0.0036
0.0155
-0.0154
0.0418
-0.0105
-0.0748
0.1413
-0.0036
0.0380
-0.0973
0.0312
-0.0014
-0.0220
-0.0142
0.0250
0.0130
0.1498
-0.0105
0.1046
-0.1325
-0.1007
-0.0145
0.0400
0.1489
-0.0279
0.2465
0.0798
0.3961
0.1511
-0.2056
0.0446
0.2661
0.4139
-0.2827
-0.6298
0.1588
0.2754
-0.3238
0.0594
-0.0599
0.0159
0.0145
0.0204
-0.0276
-0.0236
-0.0457
-0.0452
0.6017
-0.1181
-0.1775
0.1471
0.0342
-0.0379
-0.0840
0.0415
-0.4550
0.0429
0.0595
0.2269
-0.0573
-0.2034
-1.1197
-1.8118
0.0013
1.9401
-0.1005
0.0353
-0.1001
0.0803
0.1385
-0.0336
0.0341
-0.0005
0.0048
-0.0165


3
size of LD(oj) provides a measure of how much the perturbation affects parameter
estimation.
Effects on a subset of parameters, 0\ from O' = (0^, 02), can be obtained by
using the profile likelihood displacement:
LDM(u>) = 2[L(Ou02-,y) L(0lu,g(0luy,y)}, (1.2)
where 0UJ is from 0'w = (d'lu), 02uJ), and g{0\J) is the vector that maximizes the profile
log-likelihood, L(0iu.02), with respect to 02. The profile likelihood displacement
allows influence to be directed through 01, with the other parameter estimates
only changing in sympathy.
In the next section it is shown how to obtain diagnostic tools from a
local influence analysis of the likelihood displacement and the profile likelihood
displacement.
1.3 Local Influence Diagnostics
Although we could simply choose values for u> and compute LD(uj), this
is not particularly useful for two reasons. First, LD( function of a;. Hence, if we wish to find its value, re-fitting the model may be
necessary. Second, even if calculating LD(lj) was easy, we must somehow decide
what values to use for u>. This choice would be highly data-dependent and a
nuisance for the analyst.
Therefore, we would like a methodology that can examine the information
contained in the function LD(u) and yet does not require a choice of actual values
for cj nor involve re-estimating the model parameters. Following Cook (1986),
this can be accomplished using the notion of curvature. Simple plots can then be
used to identify whether estimation is sensitive to any particular elements of the
g-dimensional perturbation.


170
(a) Composite residuals
(b) Residuals
10
7.5
5
2.5 ,
2 4
2.5
-5

6 8 if K

-7.5
-10

(c) Residuals and composite residu- (d) County BLUPs
als
Figure 5.1: Corn data; diagnostic plots


132




0

# 9
9 9


h
%
rV" :
j i ,
* h;> 4.
§l

1
I



-8 -6 -4-20 2
h
Figure 4.6: Leaf blotch data; Model II; Pearson residuals vs. linear predictor
Figure 4.7: Leaf blotch data; Model II; vmax. The first ten indices correspond to site
1, varieties 1-10, while the second ten indices correspond to site 2, varieties 1-10 etc.
The dotted vertical lines are present to help differentiate sites and varieties within
sites.


This eigensystem leads to a diagnostic plot for identifying specific elements that
are influential, and it is shown how the results relate to principal components,
collinearity, and influence diagnostics.
Next, it is shown that the conceptual framework and computational tools
easily extend from maximum likelihood to inference that is performed using
estimating equations. In particular, local assessment of parameter estimates and
standard errors derived from optimal estimating functions is addressed. Algebraic
expressions and computational formulas are given for both first- and second-order
assessment, and applications such as quasi-likelihood are considered.
Finally, the techniques are applied to linear mixed models in order to assess
the influence of individual observations and clusters. While previous applications
considered only estimation, the effects of perturbations on both estimation and
prediction is addressed. It is shown that the effects on the fixed and random effect
inferences can be assessed simultaneously by using an influence measure based on
the unstandardized predictive log-likelihood of the random effects.
Vlll


183
(a) ^max for Po
(b) |vmax| for Po star-plot
(c) ^max for Pc,
(d) l^maxl for Pcom Star-plot
1
0.75
0.5
0.25
-0.25
-0.5
-0.75
-1

J 6
10 12
(e) ^max for ps
(f) l^maxl for ¡3soy star-plot
Figure 5.10: Corn data; Scheme 3; Umaxs for individual BLUEs


122
Now the elements of s7 can be constructed:
dm(u), 0W)
d/3i
.d3L{0\ y)
0 =tracc( 3/3,3930'
= trace(
= trace(
= 0.
0 jtX'Xi
hX\X
^(X'X)-1
0
O'
2<74
n
0
X'Xi
n *
X'X(X'X)-1 0
Also,
dm(u>, 0U)
da2
trace(
= trace(
= trace(
p + 3
d*L(0;y)
da2dOdO'
Mx'x) 0
O' is
i-h'1))
^(X'X)-1 0
O' *2_
n
4/ o
o' 4
)
(4.66)
(4.67)
Hence, all but the last element of s7 is zero:
0


178
(c) Umax for $c,
(d) |umax| for (3COrn star-plot
1
0.75
0.25
-0.25
-0.5
-0.75

-1
(e) V max for 0soy
Figure 5.6: Corn data; Scheme 2;
lr
-1-
(f) l^maxl for Psoy star-plot
Umaxs for individual BLUEs


7
Figure 1.4: Intersection of plane and surface creates space curve
where the vector v determines the direction and the value a determines the position
relative to u;0. Let Pv denote a plane that contains u>v(a) and extends directly
upward from 2 to intersect LD(u>) (Figure 1.4). The intersection of Pv and LD(u>)
forms a space curve, (cvv(a)', LD(u;v(a)))1 Because this space curve is contained
within the plane Pv, its curvature can be obtained as if it were a plane curve,
(a, LD(u:v(a)). Thus the curvature of LD(ojv(a)) at cj0 is simply
Cv
d2LD(uv(a))
da2
a=0
(1.9)
Different directions produce curvatures of various sizes. The normalized vector that
produces the largest curvature is denoted vmax and is of particular interest.
1.3.3 Direction of Maximum Curvature, vmax
The relative sizes of the elements of vmax indicate which elements of u> are
most influential on the parameter estimates. For example, if the ith element of vmax
1 More formally, these space curves are called normal sections. The term comes
from the fact that each plane, Pv, contains the surfaces principal normal vector, the
vector that is orthogonal to the surfaces tangent plane at u;0. Curvatures of the
normal sections are referred to as normal curvatures.


47
A similar measure can be obtained by examining local influence on
= x'iP when using the ith basic variance perturbation. That is, only the
ith observation is reweighted, and the curvature for directed influence on is
examined. The curvature for the ith basic perturbation is the ith diagonal element of
jj,**(*<).
Cbi = BasicFITS* = (2.32)
A large value of BasicFITS indicates that the perturbation of the observations
precision/weighting greatly influences its fitted value. Thus it is a local influence
analog of DFFITS. Coincidently, BasicFITS* is equivalent to the ith basic curvature
for the acceleration matrix, as given in (2.20). That implies that for the variance
perturbation, the n basic curvatures of LD(uj) are measuring influence on the n
fitted values.
While BasicFITSj is the ith basic curvature when directing influence on /2(x*),
we can also define MaxFITS* to be Cmax for directed influence on fi(xi). Using
(2.31),
Cmax. = MaxFITSj =
a2h
zY,hlrl
(* j=i
(2.33)
By definition the, ith MaxFITS can be no smaller than the ith BasicFITS. While
BasicFITS provides information about which observations perturbation influences
its fitted value, MaxFITS provides information about how sensitive each fitted value
is relative to the others.
Example
Consider examining influence on the fitted values for each of the 35 covariate
profiles in the Scottish hills data set. Figure 2.2(a) displays the 35 BasicFITS (i.e.,
each of the basic curvatures for directed influence on the corresponding /i(x)) .
Race 7 is indicated as being influential on its fitted value, while race 18 is also shown


148
case weight was introduced as a perturbation. Because the perturbation does not
correspond to a new density for the data, it would be considered a perturbation of
the estimation technique. Diagnostics were primarily based on basic perturbations
(i.e., one-at-a-time perturbations) of subject weights. Lesaffre and Verbeke (1998)
showed that the basic curvatures could be written as the sum of several terms, and
the easily-interpretable terms were used as a means to compare the relative influence
of subjects.
5.5.2 Assessment of Predictions
The influence of perturbations on both the BLUEs and BLUPs can be
assessed by applying the techniques developed in Chapter 4 to the predictive
likelihood displacement (5.7). The estimating functions, h(0;y), are those used
in the mixed model equations (5.18), and if the perturbed model is also a linear
mixed model, the perturbed estimating functions, hiJ(0;y)1 are also of this form.
Local assessment of the predictive likelihood displacement requires the following
quantities:
7 (0) = 0U
0 =
u
m = PLD(u) = 2[PL(] y, ¡3) PL{uu-y, 0U)].
The next step is to determine whether first- or second-order local assessment is
appropriate. Since the influence measure does not explicitly depend on w, = 0.
Also, by derivation of the mixed model equations,
dPLD{ u?)
dl{0j) o
o


52
3.2.2 Directed Influence on a2
The acceleration matrix for the profile likelihood displacement for a2 is given
by
4 4 0}
na
rr'.
(3.6)
(3.7)
Solving the characteristic equations (F XI) v 0 directly, the single
eigensolution is
Cn
^max ^ ^
Therefore the direction of maximum curvature for directed influence on a2 is always
proportional to the residuals regardless of which column of X is perturbed, but the
corresponding curvature is a dependent upon the effect size, \$i\.
3.2.3 Directed Influence on &
Using the results of Beckman et al. (1987), the sole eigenvector is proportional
to A'T and the corresponding curvature is 2||A'T||2 as per (2.3):
1
cr
i
^max ~Z
a
r PiX i P\X(\)
IM
7i
1
Ikill
i
Ikill
r-Xi+M 1*1
r fari
(3.8)
oc r /fixi


94
around the null perturbation:
r\
m(a) ra(0) + (a 0)
da
= rri(0) + (4-14)
OJ1
ra( 0) + arh'v,
where the partial derivatives are evaluated at a = 0. The quantity rh'v has a
geometric interpretation. Consider the influence graph of m (i.e., the influence
function described as a surface in 3i9+1). For each direction v, there is an associated
lifted line on the influence graph. The slope of this slice of the graph is given by
S = rh'v, (4.15)
and it is sometimes called the directional derivative. The direction with the largest
absolute slope is proportional to rh, the gradient vector.
Both the directional derivative and the gradient vector have diagnostic value.
To explain, recall from Chapter 2 that we may use a perturbations Euclidean
distance from the null perturbation, ||w w0||, as a measure of its size. Further, let
us define the effect of the perturbation as |m(a) ra(0)|. If m(0) = 0, the effect of
a perturbation is the absolute value of m(a). The directional derivative can then
be interpreted as the approximate effect of a size-1 perturbation in direction v.
Furthermore, suppose that we have perturbations in two different directions:
in direction and cjg in direction vB- If \S\ is k times larger than \Sb\, then
perturbation umust be k times larger than to have the same effect. Therefore,
directional derivatives are measures of the relative influence of perturbations.
Additionally, the gradient vector shows how perturbations can be scaled to achieve
maximum local influence. That is, w0 (m/||m||) has the largest approximate
effect, ||m||2, among perturbations of size 1.


53
and
Cmax
<72||ri||2
|r Anil5
ri
r'r 2Arir' + $l\\ri\
_A/>+Hi
v2 Vft Ikill2
(3.9)
The direction of maximum curvature is a weighted average of the residuals and the
adjusted regressor. As noted in Cook and Weisberg (1991), fari = y y^, where
j/(i) is the vector of fitted values when X\ is excluded from the model. Using this
result, we find umax a 2r r*^, where is the vector of residuals when modeling
y using all regressors except Xx. This indicates that the direction is dominated by
the residuals, a fact which is demonstrated by examples later in this chapter. Cook
and Weisberg (1991) discussed how umax relates to de-trended added variable plots.
3.2.4 Directed Influence on Other Regression Parameter Estimates
Thomas and Cook (1989) examined the influence that perturbations in one
column of X can have on the regression parameter estimate corresponding to a
different column. Without loss of generality, suppose we perturb the first column of
X and look at the effects on the last regression parameter estimate /3fc. Again using
the results of Beckman et al. (1987) we can obtain the maximum curvature and its
direction. Letting d\ be of size A: 1,
oc A'T =
(r< AX(t>) -$Xt
a
~7k
J INI
1
Ikfell l
ocAi r + fark,
rd'i7 k + PiH(k)Xk AXfe
(3.10)


152
The building blocks for deriving h are as follows:
lr
X = ln
z =
'm
G = R = a2 In
XX = n Z'Z = D(nt)
X'Z = n' = (ni,n2,.. .ng).
It follows that
-h =
X'RxX X'R'Z
Z'R~1X Z'RlZ + Gl
a *11
¡n jsD(m) + £lg
n
n D{
n
<72+n£7
)
(5.33)
The following results for a partitioned matrix with a scaler upper-left sub-matrix
can be used to invert the matrix (Searle 1982):
A B
1
C D
c
1
-BD
+
0 0
0 D1
-D~lC D lCBD l
(5.34)


18
Influence on both regression parameter estimates
Figure 1.10(e) displays the direction of maximum curvature for directed
influence on 3- Plot (f) is the direction of largest curvature under the constraint of
orthogonality to t>max. The two plots are very similar to the directed influence plots
in (c) and (d).
1.5 Basic Perturbations
As an alternative to finding the direction of maximum curvature, we may
utilize basic perturbations. The ith basic perturbation is the same as the null
perturbation except for an additive change of 1 in its ith element. For example, the
first basic perturbation is u>o + (1,0,..., 0)', and corresponds to moving a distance
of 1 along the first axis in fl. Typically, the curvatures for each of the q basic
perturbations would be computed and then plotted together.
As an example, consider applying the variance perturbation to a linear
regression, and examining directed influence on 3- In this case, the ith basic
curvature, C^, is a local influence analog to the ith Cooks Distance, because it
corresponds to re-weighting a single observation. It is shown in Section 2.7.2.1 that
the ith basic perturbation for this scheme measures influence on the ith fitted value.
Figure 1.12(a) plots the 35 curvatures for the basic perturbations applied to the
Scottish hills data, while Figure 1.12(b) plots the 35 Cooks Distances. Although
the scales differ, they provide similar information. Both plots draw attention to race
7, but they differ somewhat in their emphasis of races 11 and 18.
1.6 Brief Literature Review
Local influence techniques have been applied to a number of statistical
models, most notably univariate and multivariate linear models (Cook 1986; Cook
and Weisberg 1991; Lawrance 1991; Schwarzmann 1991; Tsai and Wu 1992; Wu
and Luo 1993a; Kim 1995; Kim 1996c; Tang and Fung 1996; Fung and Tang 1997;


To Christy


51
The next step is the construction of A. Partial derivatives with respect to w are as
follows:
= ft ( X|8 Ah,)
OW <72
Second partial derivatives are then
^^>=i(-X(1)0(1)-2A(Xl+w))
&Lu(e.y) 0lv
dvd/31^ a2 !1
(74
dwd(72
Evaluating at the original model yields
82K(9:y)
dudP[
d2Lu{0-y)
1
dudf3'{l)
&Lu{0-y)
= -(r-/3lXl)
0=0, (jj= = ~^2 A(l)
0=0,U=Wo a
duda2
Pi
0=0,U=Uo
and from these quantities we construct the n x p matrix
A' =
a2
(3-4)
r-hXx -PiX(l) -§*i
rd[ $xX -%r
where dx is a k x 1 vector. It follows that the acceleration matrix is given by
F = 2
rd[-diX -hr
o2 (XX')"1
0
1
dxr' P\X'
0'
2l74
n
a2
-4 r'
+ |?% ) rr' + plH -J^(rr\ + rxr')
kill2 Ik112
ri
(3.5)


169
plot indicates that observation 33 is the most outlying from the overall estimated
regression curve, X¡3. Figure 5.1(b) shows the residuals e = y X(3 Zu. This
shows that even when taking into account the county effects, observation 33 is
the most outlying. Figure 5.1(c) shows both sets of residuals together, with the
composite residuals represented by stars, and the residuals represented by triangles.
We note that the largest difference among them is for counties 5 and 11, where the
composite residuals are nearly all smaller. Not surprisingly, counties 5 and 11 have
the largest BLUPs in Figure 5.1(d). Also, because the residual for case 33 is large
and a different sign than the other residuals for county 12, it appears that i2 would
be considerably larger if not for observation 33. A local influence analysis can assess
this possibility more thoroughly.
5.9.3 Scheme 1: Residual Variance Perturbation
Figure 5.2 contains local influence diagnostics for perturbing the error
variance in the corn model. The first plot shows the basic curvatures, Cb, that
result from one-at-a-time perturbations. Observation 33 is considerably more
influential than the others owing to its noticeably large curvature. The element
for case 33 dominates the direction of largest curvature (Figure 5.2(b)). If all
elements were equally contributing, then they would equal either 1 = ^ = 164 or
.164. Therefore, the value for element 33 is over five times larger than the value
of equally-contributing-elements (e.c.e.). Plots (c), (d), and (e) are the directions of
maximum curvature for directed influence on the intercept, the CORN slope and the
SOY slope respectively. The plots indicate that case 33 is a dominate contributor
to the SOY slope in particular. Plot (f) displays the largest curvatures for directing
influence on the 12 predicted random effects. This plot shows which BLUPs are
most sensitive to perturbation. Here we see that the county 12s BLUP is the most
sensitive to the case-weights. This is the county containing the questionable data
point.


TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iv
ABSTRACT vii
CHAPTERS
1 INTRODUCTION TO LOCAL INFLUENCE 1
1.1 Premise 1
1.2 Measuring the Effects of Perturbations 1
1.3 Local Influence Diagnostics 3
1.4 Perturbations in Regression 8
1.5 Basic Perturbations 18
1.6 Brief Literature Review 18
1.7 Outline 20
2 COMPUTATIONAL ISSUES IN LOCAL INFLUENCE ANALYSIS 21
2.1 Curvatures and the Direction of Maximum Curvature .... 21
2.2 Local Influence Analysis of a Profile Likelihood Displacement 23
2.3 Building Blocks of Local Influence for Regression 24
2.4 Perturbations to the Response in Regression 27
2.5 Unequal Measurement Precision Applied to Regression ... 31
2.6 Benchmarks for Perturbation Analyses 36
2.7 Local Assessment Under Reparameterizations 40
3 PERTURBATIONS TO THE X MATRIX IN REGRESSION ... 49
3.1 Perturbations to Explanatory Variables 49
3.2 Perturbations to a Single Column 50
3.3 Perturbations to a Single Row 59
3.4 Perturbations to the Entire Design Matrix 65
4 LOCAL ASSESSMENT FOR OPTIMAL ESTIMATING FUNCTIONS 87
4.1 Inferential Platform 88
4.2 Local Assessment of Perturbations 93
4.3 Acceleration, Curvature and Normal Curvature 96
4.4 Algebraic Expressions for Local Assessment 98
4.5 Computational Formulas for J and P 103
v


124
1
0.5
x
I 0
-0.5
-1
Figure 4.1: Hills data; umax for perturbed log generalized variance of 6
variance of these two estimates is considered separately, the directions of maximum
absolute slope are respectively given by
f max OC TSq (7 ln X)max OC Vaq
The direction of maximum slope for p,u is dominated by squared residuals that
deviate greatly from the average squared residual, by large squared residuals. The direction of maximum slope for the joint influence
measure is simply a weighted average of these two vectors.
4.7.2.5 Examples
As examples, we consider the Scottish hills data and the school data. For
the hills data, the element of umax for observation 11 is large and the element for
observation 18 is huge (Figure 4.1). For the school data, elements 3 and 18 show
substantial influence on the perturbed generalized variance (Figure 4.2).
15 20
Case


46
2.7.2 Application to Regression
As an example, consider perturbing the variances of a regression model and
examining influence on a linear combination of the regression parameter estimates,
fi(x) = x'¡3. From (2.28), we find
Frt') = jjApA 'll(d1(X'X)-')xx'(d1(X'X)-1)Ail
a2ht
D{r)X(XlX)-lxx\X'X)-lX'D{r),
(2.29)
where h* = x'(X'X) 1x Further,
Umax oc x'(X'X) 1X'D(r) oc Cov(X¡3,x'¡3) r
2
Cmax
\x' (X' X)~l X' D(r)\\*,
(2.30)
(2.31)
max a2K
where Cov(X¡3,x'¡3) is the vector of empirical covariances between X¡3 and x'¡3.
2.7.2.1 A DFFITS-type measure
Using the variance perturbation and directed influence on x'(3, a local
influence analog of DFFITS can be defined. Recall that DFFITSj is defined as the
externally studentized change in the ith fitted value when deleting the ith observation
(Belsley et al. 1980):
DFFITSj =
\Js[i\hii
where fi(xi)]^ is the estimate of /x(ajj) with the ith observation deleted and is the
mean squared-error estimate of cr2 with the ith observation deleted. A large value
for DFFITS indicates that the observation is somewhat self-predicting. It can be
shown that DFFITSj simplifies to
r.hV2
' *",ii
(1 hn)./sl
M


Copyright 2000
by
Glen Lawson Hartless


23
In these situations it may be more prudent to look at directed influence on single
parameter estimates.
2.2 Local Influence Analysis of a Profile Likelihood Displacement
The profile likelihood displacement, LD^(oj), was presented in Section
1.2 as a measure of influence directed through G\. Cook (1986) showed that the
acceleration matrix can be expressed as
F[6l] = -2A'(X"1 B22)A, (2.2)
where
B
22
0 0
o -^22
and L22 comes from the partition
where
d2L(Q\ y)
lJ dOM
9=9
This result is proved in Chapter 4 as a special case of Theorem 4.2.
Beckman et al. (1987) considered directing influence on a single parameter
estimate. They noted that in this case, the acceleration matrix is the outer product
of a vector, and hence has only one non-zero eigenvalue. Without loss of generality,
suppose that we wish to direct influence on the first parameter estimate in 0. The
eigensystem is then given by
v oc A'T,
A = 2|| A'T
(2.3)


6
Figure 1.3: Likelihood displacement with directional vectors
where g(x0) is the functions minimum. This result can be applied to LD(uj) at u>o
when there is only a single perturbation, u. Here, the curvature is given by
d2LD(uj)
dui2
(1.7)
UJUJo
and it measures how quickly the parameter estimates change when uj is moved
away from the null perturbation. A large curvature indicates that even if u is close
to, or local to u>0, the likelihood displacement increases substantially. Hence, a
large curvature indicates that even a slight perturbation can be very influential on
parameter estimates.
1.3.2 Using Plane Curves to Describe LD(u>)
The curvature of plane curves can also be used to describe LD{w) when
q > 1. Now u> can be moved away from w0 in different directions, as shown by
the vectors in Figure 1.3 for a two-dimensional perturbation. These vectors can be
written as
cov(a) cv0 + av
(1,8)


160
as well the correlations among observations in the same cluster. Hence, the only
effect of the correlation is to change the relative weighting of the clusters, allowing
us to assess the influence of clusters of observations.
5.8.2 Computational Building Blocks
To derive A we begin with the perturbed joint likelihood:
Lu(u, (3; y) = log{f(y | /3, u)g(u | /?))
-![(,- x-ey RZ1 (V x0) + u'g-Ju]
= [v'K'v 2 ffx-Rz'v + e'x-'Rz'x-e + ' Partial derivatives with respect to the fixed effects (3 are:
dL(u,0\y) = _1 [_2 X'Rz'y + 2X'RZlX0 + 2X'Rz'Zu\
= XRZ'(V -X0- Zu)
a
= ^im
partial derivatives with respect to u are:
dL(n,/3;y)
du
= Z'R~1(yi Xif3 ZiUi) G-Ju
9
= Y, i=l
The perturbed estimating functions are then
M#; v)
ZLlUiX'iR-Hyi-xi0-z,ui)
uZZ[R-l(y, Xi@ ZiUi) GT'ui)
(5.41)


139
5.3.1 Predictive Likelihood Displacement
An overall measure of the change in estimates (3 and predictions due to
perturbation is the predictive likelihood displacement:
PLD(u)) = 2[PL(u- y, 0) PL(ww; y, 0U)\. (5.7)
A local assessment of this influence measure could be first- or second-order,
depending on the model and the method of inference.
Thomas and Cook (1990) considered influence on predictions in generalized
linear models (GLMs). To explain, let u denote a single future observation that
is independent of the observed data. Also, let /( \0) denote the density function
shared by the observed data and the future observation. Thomas and Cook (1990)
proposed the following influence measure for assessing the effects of perturbations
on predicting u:
d*{u) = log/(£|/3) log/(u|/3J. (5.8)
In the context of the article, parameters were estimated by ML and point predictions
were = E(u|y,/3). For ML estimation of a GLM, d*(u) is non-negative and
minimized at the null perturbation, indicating that a second-order local assessment
is appropriate. However, in other situations it can be negative.
The influence measure d*(w) is motivated by a deviance residual for the ith
observation (McCullagh and Nelder 1989):
\ogf{yi\pi = Vi) log/(y|y = /}(/3)), (5.9)
where E(y) = and /i is the estimated mean of the observation based on the
model. In the context of GLMs, each deviance residual is non-negative, with smaller
values indicating a better fit. The deviance of a model is defined as the sum of the n
deviance residuals and serves as measure of overall goodness-of-fit.


123
4.7.2.3 Directional derivative
Using the results above and A from Section 2.5, we may finally construct the
directional derivative of the log of the perturbed generalized variance
S=(8'u + 8'KJ)v
= (8'u + 8'K(-L~ )A)v
= K +
= ,+
O' 2#
q/ (p+3) a^X'X)-1 0
0'
2&4
n
j*X'D(r)
J_r<
2 a4 sq
)v
. p + 3 .
= ( +
ncr*
A)v
2 | ^
= (-[ D(Xx) ... D(Xp_t) ] [Ip-1 X] vec(X'X)-1 rsq + ^-rtq)'v
na
= (-[ D(Xx) ... D(Xp_x) ] [IP-1 X] vec(X'X) 1 + ^-rtq)'v.
I t(J
(4.68)
Hence, the direction of the maximum absolute slope is
Umax OC -[ D(Xx) ... D(Xp-i) } [Ip-1 X] vec(X'X)-1 + ^4^rsq. (4.69)
Tb0"
4.7.2.4 Simple random sample
As a special case, consider a normally distributed simple random sample
(SRS). The direction of maximum slope simplifies to
1 3
Umax OC m = -D(ln) [1 (8> 1] vec(-) + ~^rsq
TI nrr
na*
oc
sq
- T1"
(4.70)
. 2
Hence, observations with squared residuals that deviate largely from have large
elements in umax. This may seem odd, but recall that this represents influence on
an estimated joint asymptotic confidence region for and a*. If the log of the


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Jams G. Booth, Chairman
Professor of Statistics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Dofctoj/of Philc
Professor of Statistics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation j*Jid is^ully^ac^equate, in scope and
quality, as a dissertation for the degree of Doc
Japes
ssociate Professor of Statistics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
X Cwfcu
Randolph L. Carter
Professor of Statistics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
3s J. Algina
fessor of Educate
1 Psychology


82
(a) |Umax| for a1
(b) |t>max| for 0 (c) |v2| for 0
Figure 3.8: Hills data; X + W pert.; column profiles of directional vectors. DIS
TANCE values are connected by solid line. CLIMB values are connected by a dashed
line.


34
This suggests that b2 is most affected by variance perturbations when they are
scaled according to the squared residuals.
2.5.3 Directed influence on /3,
Without loss of generality, consider directed influence on Again we will
proceed directly to determining the sole eigenvector proportional to A'T and the
corresponding curvature 2||A'T||2 as per (2.3) and (2.9):
oc ^rD(r)
Xl X(1)
1
lln
D(r) [X, X(1)7,]
D{r)r\
1
-7i
(2.17)
oc r Vi
and
Cmax = 2||-
O Fi
-r t*!||2 = 2n-
\r n
r||2||ri||2
(2.18)
This suggests that maximum local influence of the perturbation on $ is
achieved by perturbing cases that have both large residuals and extreme values for
the adjusted covariate. Cook (1986) provided additional details of how this result
compares to the information in a detrended added-variable plot.
Also, we may compare the information in umax with that provided by the
DFBETAS diagnostic (Belsley et al. 1980). Recall that the value of DFBETASij
is the standardized difference between fij and the estimate when deleting the
ith observation. The formula is given by
DFBETAS^ =
@3 PjAi)
\Jvar0j)
C-ijTi
sllCjll(l ^t)


137
A predictive likelihood for u would result by somehow eliminating 0 from the above
expression. For Bayesians, this can be achieved in a straightforward manner. The
posterior density of 0 given the data y is obtained, and then (5.6) is integrated over
the density of 0. The resulting expression is a posterior predictive likelihood for u.
For frequentists there are several competing options, as outlined in Butler (1986)
and Bjprnstad (1990). Here we consider two: the estimative predictive likelihood
and the profile predictive likelihood.
5.2.1 Estimative and Profile Predictive Likelihood
The estimative/empirical predictive likelihood replaces 0 in (5.6) by 0, its
MLE. The resulting predictive likelihood is actually proportional to the density of u
given y with the MLE inserted in place of 0. This is considered a naive approach, as
resulting inferences for u do not take into account the uncertainty in the estimated
parameters. An alternative frequentist predictive likelihood can be motivated by
profile likelihood. The idea works as follows. If 0 is considered a set of nuisance
parameters, then a technique for estimation is to first maximize the joint likelihood
of (u,0) with respect to 0 for fixed u. The resulting solution, 0(y,u), is then
substituted into the likelihood. The resulting expression is a function of only u and
y, and is known as the profile predictive likelihood of u (Mathiasen 1979; Lejeune
and Faulkenberry 1982; Levy and Perng 1984).
5.2.2 Point and Interval Predictions
Regardless of which predictive likelihood is chosen, point and interval
predictions are of interest. One option for a point prediction is to maximize the
predictive likelihood, £(u] y) with respect to u. However, the resulting maximum
likelihood predictor can have poor properties (see Bjprnstad, 1990, examples 4 and
7). An alternative can be motivated by standardizing the predictive likelihood to be
a predictive density for u: f(u\y) = £(u; y)//2(y). The mean of this conditional


158
We now consider influence on the predicted random effects. Without loss of
generality, let us examine influence on \. The direction of largest local influence is
given by
u OC A'h do
-1
a D(e) [X Z]
a2 i k
<72+ o-|+o-2/n9
= D(e)
-
"
Ini
1
1
M
3
CQ
1
-1
a2 i k
cr2+a2 /ng
= D(e)
-1 +
a2
t2 4-/t2 !
!
oJ+.a21 n\ nj
-1 +
1 I
-1 + aj+^rg
(
_ ni <72+<72/ni
^Z"2 e
o,|-fa2/n2
)Cl
(DU ?
Z2¡
|+ (T2/t12
)el
(5.40)


20
and Poon 1999). Extentions to Bayesian methodology have also been considered
(McCulloch 1989; Lavine 1992; Pan et al. 1996).
1.7 Outline
In Chapter 2, computational formulas for local influence analysis are discussed
and derived for the two perturbation schemes discussed in this chapter. Chapter 3
contains derivations and examples of local influence diagnostics for linear regression
under perturbations to the X matrix. In Chapter 4, the methodology is extended to
a general class of influence measures and estimation techniques. Computational tools
are also provided. In Chapter 5, inference and local influence analysis for prediction
problems are discussed. Techniques for assessing estimation and prediction in linear
mixed models are developed and applied to data. In Chapter 6, some preliminary
results that can be applied to multi-stage estimations are presented. The chapter
also contains commentary on additional areas of future research. The appendices
contain details for data analyses.


71
The quantity R2 is the proportion of variance explained by the model, and serves
as a measure of goodness-of-fit. However, R2 can be made artifically large by
adding additional explanatory variables, even when they contribute very little to the
explanatory value of the model. The quantity ||/3||2/(J2 behaves similarly: additional
variables decrease the denominator and usually increase the numerator. Of course,
the scale of the regressors is a major factor in the size of ||/3||2, indicating that
scaling the regressors would be prudent for interpretation.
So, we may interpret Cmax in the following way. The vector of parameter
estimates is more sensitive when (a) there are collinear variables, (b) there are a large
number of explanatory variables, and (c) the model closely fits the data. We are not
surprised to see that the first two phenomenon are sources of sensitivity. However,
the suggestion that better-fitting models are more sensitive to measurement error
may at first be disconcerting. When viewed in light of Type I and Type II errors,
this is not the case. Recall that a Type I error occurs when a test is wrongly
declared significant, while a Type II error occurs when a test is wrongly declared
insignificant. Generally, a Type I error is considered the worse of the two. So, if
there are strong relationships between X and y, the formula for Cm?ai implies this
might be missed because of measurement error in the regressors. However, if there
is a weak relationship, perturbations to X would not make the relationship appear
strong. In this sense, the regression model is better protected against Type I errors
(concluding there are strong relationships when there are none) than Type II errors
(concluding there are weak relationships when in fact there are strong ones) when
there is measurment error in the regressors.
As a final comment on Cmax, the term ||/3||2/ an indicator of a closely fitting model rather than a well fitting model. This suggests
that if we construct complicated models to fit the nuances of a data set, the resulting


173
(a) timax for ui
(b) Umax for 2
(c) Umax for 3
(d) Umax for 4
(e) Umax for 5
(f) Umax for 6
1
0.75
0.5
0.25
0.25
-0.5
0.75
-1
!
> *5V 10
0.75
4
J
0.5
0.25
A
4 i 4
(g) Umax for U7
0.25
0.5
0.75
1
(h) Umax for 8
0.75
0.5
0.25
0.25
0.5
0.75
1
0) ^max for 10
(k) Umax for 11
(1) Umax for 12
Figure 5.3: Corn data; Scheme 1; directed influence on individual BLUPs


125
1
0.5
-0.5
-1
0
5
10
Case
15
20
Figure 4.2: School data; t;max fr perturbed log generalized variance 6
4.7.2.6 A simpler measure
The directional derivative simplifies greatly if the measure is based on the
unperturbed generalized variance evaluated at the perturbed estimates:
In this situation, su vanishes, but the other derivations follow from above, yielding
the following results for the variance perturbation:
12 max OC VSq.
4.8 Local Assessment for Quasi-likelihood Inference
In this section, local influence assessment of quasi-likelihood estimates
(Wedderburn 1974) for independent observations is outlined. A thorough treatment
of quasi-likelihood inference can be found in McCullagh and Nelder (1989). Here, we
present only some fundamental concepts, viewed from the vantage point of optimal
estimating equations (Godambe and Heyde 1987; Desmond 1997a). Some local
influence techniques are then outlined.


40
parameters 0. The expected curvatures can then be plotted along with the observed
curvatures (Cadigan and Farrell 1999). If the expectation of Cmax is difficult to
obtain, Monte Carlo simulations could be used to estimate it. As an alternative,
Schall and Dunne (1992) developed a means of external benchmarking by developing
a scaled curvature using results on parameter orthogonality (Cox and Reid 1987).
2.6.2.5 Internal versus external benchmarks
Internal and external benchmarks are complimentary tools, since each one
provides different diagnostic information. Comparing curvatures from the same
fitted model highlights relative sensitivity given the data and the design. Comparing
curvatures to their expectations highlights whether the sensitivities are expected
given the experimental or sampling design.
2.6.3 Calibration of Directional Vectors
For a directional vector from the eigensystem of F, the analyst must compare
the individual elements in the vector to determine which aspects of the perturbation
are most important. Typically, it is suggested that the analyst simply look for
elements that stand out as being large relative to the rest. Alternately, a vector
whose elements are all equal in size can serve as a means of comparison. That is to
say, if each of the q elements of the perturbation are contributing equally, then each
is equal to either or -T. A particular element whose absolute value is larger
than, say, 2 or 3 times that of equally contributing elements (e.c.e.) would then be
of interest. Hence, the inclusion of horizontal lines at -^= and ^= in the plot of
directional vectors may be useful.
2.7 Local Assessment Under Reparameterizations
Pan et al. (1997) show that vmax is invariant to any 1-to-l measureable
transformation of the parameter space 0. In fact, for 0 /i(0), the entire
eigensystem of F$ is the same as that of Fg. However, directed influence on the


165
The acceleration matrix is then
-i
M = -2A'h A
-2 0 -D(u)'G
22
-2D{u)'G~lh G lD{u)
11
12
h
h
0
h21
h22
-GlD(u)
(5.49)
= 2D(u)'G~l {[Z'R~lZ + G-1)-1
+ GZ'V-lX(X'V-xX)-lXV-lZG\ G~lD{u).
This matrix has at most p2 distinct non-zero eigenvalues regardless of the number of
fixed effects. In addition, using the results given in Section 5.5.3, the acceleration
matrix for influence measure PLD^(uj) is equivalent to (5.49). Finally, the
acceleration matrix for PLD^\uj) is
i /ti/ 0 0
- )
(5.50)
= 2D(u)'ZlV-1X{X'V-1X)-lXV-1ZD(u).
As special case, consider the one way random effects model. Using Theorem
2.1, the direction of maximum curvature for ¡1 is
, -1
1
o
^T
i
)
0
M =
0 -D(u)G_1
(/i -
0 0
-G~lD(u)
-i
Umax 0^ ^ h di
OC
0 D(u)
o-2+o-2/ni
a2+a2/ng
tr2+a2¡n\
Zin
. o-2+ (5.51)


Examining the second term of d2m(6)/du)du/ and evaluating at the null
perturbation, we find
103
fdm(S)
by Lemma 4.1.
[< 9 iq\ r
[s\ Iq] ([Jr J'] GJ + [K Iq] P)
[; J'] GJ + [s\K Iq\ P
4.4.3 Comments
While the algebraic expression for S is quite simple, the expression for A
is somewhat cumbersome. However, a second-order assessment would typically be
used only when the gradient vector vanishes. In most situations then, sT = 0, and
the acceleration matrix simplifies to
M =
Iq J'K'
S
I*
KJ
(4.32)
The algebraic expressions are helpful in that they decompose the diagnostics
into building blocks, most of which are immediately computable. Specifically, the
quantities su, s7, S, G, and K should be immediate upon choosing an influence
measure and fitting the model. In the next section, it is shown that the matrices J
and P are also easily obtained in most situations.
4.5 Computational Formulas for J and P
In order for the formula for the gradient and acceleration matrix to be of
practical use, they must be easily computed in a wide range of applications. In this
section, it is shown that the matrices J and P can be computed even when the
unperturbed parameter estimates are not closed-form. The only requirement is that
the perturbed estimating functions be closed-form expressions of u.


131
Figure 4.4: Leaf blotch data; Model I; vmax. The first ten indices correspond to site
1, varieties 1-10, while the second ten indices correpsond to site 2, varieties 1-10 etc.
The dotted vertical lines are present to help differentiate sites and varieties within
sites.





N fc'J** a.* #


1
% ..


y .

-8 -6 -4-20 2
h
Figure 4.5: Leaf blotch data; Model I; vmax vs. linear predictor


130
3
2
v 1
H 1
(II
0)
u
0
-1
Figure 4.3: Leaf blotch data; Model I; Pearson residuals vs. linear predictor
A plot of Umax indicates that only observation 24 is influential for this model (4.7).
The elements have less association with the linear predictor than the binomial model
(Figure 4.8), indicating that the contribution of the observations to inference is more
appropriate in the second model.





%* ,1
, ,. *? t.
1
.
%
Hi* j .i l
5V

' i



.


-8 -6 -4-2 0 2
h


128
4.8.2.3 Example: Case-weights Applied to Leaf Blotch Data
As an example, we apply the case-weight perturbation to a data set
concerning leaf blotch (Wedderburn 1974; McCullagh and Nelder 1989). The
percentage of leaf area affected by the blotch was recorded for 10 varieties of barley
grown at 9 sites. Although the response is continuous, it can be analyzed as if arising
from binomial sampling using the logit link function and a main effects model:
V(fXi) = Hi( 1 Mi) 9Mi) = log = r]i.
i Mi
Plotting the Pearson residuals from this model against the linear predictor indicates
that the model overestimates the variance for the largest and smallest proportions
(Figure 4.3). Applying the case weight perturbation requires h 1 and the following
matrix:
A = ^X'D(y A),
where X is the model matrix and D(y fi) is a diagonal matrix of the raw residuals.
To measure the effects of perturbation, we use the unperturbed Pearson Chi-squared
statistic evaluated at the perturbed MLEs:
m
V(£,J '
1=1
This implies that the final building block for a first-order assesment of this measure
is given by
Sy
dm
86
= X'zu
where the n x 1 vector Z\ has ith element
^ViMi + vK1 ~ % Mi) ~ Mi
Ai(l Mi)


16
(a) 'Umax
for 0
(b) umax for a1
(c) ^max for ^climb (d) ^max for ^dist
(e) ^max for ¡3 (f) U2 for ¡3
Figure 1.10: Hills data; variance perturbation


Row
Column
| for 0
Figure 3.13: School data; X + W pert.; bubble plot of |Wmax


LD
1780
20 M
.H233
UNIVERSITY OF FLORIDA
3 1262 08555 1843


61
Since the null perturbation is 0, evaluating at w0 simplifies the expression. We then
proceed to take second partial derivatives:
d2K(0;y)
dud fa
U=U0
0 )
0
2/i x'iP
~Xi iP
0
0 J
d2Lu(e]y)
duda2
Vi ~ xiP
P-
U=U0
Evaluating at 0 6 and combining the portions for the regression parameter
estimates we obtain
d2L {0-y)
dudp'
d2Lu(0-y)
1
O=0,U=Uo
r\I px[
duda2
= ~y4P,
0=b,U=Uo a
where rx is the first residual. From these quantities we can construct the k x p
matrix
A' =
<72
The acceleration matrix is then given by
F =
c
2
rxI-px i -£P
(3.15)
/\ o
az
rj Px[ -$P
-i
a2 (XX')"1 0
i
i
1
O' 2&1
n
T~
a
-0
a2
2r2 \
hu + 7r-¡ )PP + r\(X'X)-1 rxPx\(X'X)~l rx(X'X)~Wp'


159
Although the individual cluster sizes play into the weights, it is clear from inspection
that the residuals from the first cluster have weights roughly g 1 times greater
than the others. In the balanced case, all = n/g, and this implies
cr2/nj a2/n
a2 + a2/rij a2/n + o2/g
^ o2 ¡n\ (g l)cr2/n
It follows that
ei
Ten
g-1 9
Thus, in the balanced case, the residuals of the first cluster have weights that are
g 1 times larger than the others. In fact, the BLUP of U\ is given by
2c/9
a2/n + a2/g
Not surprisingly, maximum influence on i is achieved primarily by perturbing the
variance of observations in cluster i.
5.8 Cluster Variance Perturbation Schemes
In this section, two cluster-based perturbation schemes are considered for
assessing how clusters of observations influence estimation and prediction.
5.8.1 A Cluster-based Case-weight Perturbation
For simplicity, we consider the model given in Lesaffre and Verbeke (1998),
(5.25), but with the diagonality-constraint for Ri dropped. To perturb the model,
we let Gi become and Ri become ^Ri for i = 1,.. .g. The perturbation of
both matrices is important: it preserves the correlations among the random effects


193
Predicted Values
HECTCORN
Predicted
SE Pred
165.7600
155.4869
9.7838
96.3200
89.3834
8.8869
76.0800
98.3267
8.1004
185.3500
170.5417
8.5361
116.4300
144.2402
7.5387
162.0800
154.4196
7.0506
152.0400
125.5822
6.7333
161.7500
156.5001
7.1277
92.8800
91.0834
7.9300
149.9400
131.2890
7.2587
64.7500
65.0934
8.7890
127.0700
141.4215
7.1954
133.5500
118.8645
7.1089
77.7000
90.7578
7.6708
206.3900
184.9299
9.1302
108.3300
118.7686
6.7585
118.1700
123.7514
7.6313
99.9600
106.1059
7.3867
140.4300
123.6154
6.2159
98.9500
91.7140
8.0456
131.0400
125.3031
7.5450
114.1200
123.9749
5.9584
100.6000
97.0015
6.3382
127.8800
139.1748
6.4405
116.9000
107.4351
5.9034
87.4100
92.8847
6.9265
93.4800
85.2028
9.2386
121.0000
138.6914
6.9531
109.9100
127.4057
8.0247
122.6600
129.0737
6.0804
104.2100
111.5808
6.1297
88.5900
89.8508
6.3707
88.5900
139.1245
8.2468
165.3500
142.4030
6.2021
104.0000
106.1154
5.7054
88.6300
75.2417
8.4331
153.7000
139.6604
6.3394
L95
095
Residual
135.2477
175.7262
10.2731
70.9994
107.7674
6.9366
81.5698
115.0837
-22.2467
152.8834
188.2000
14.8083
128.6451
159.8353
-27.8102
139.8343
169.0049
7.6604
111.6532
139.5112
26.4578
141.7553
171.2449
5.2499
74.6789
107.4879
1.7966
116.2733
146.3048
18.6510
46.9121
83.2748
-0.3434
126.5367
156.3062
-14.3515
104.1587
133.5704
14.6855
74.8896
106.6260
-13.0578
166.0426
203.8172
21.4601
104.7876
132.7496
-10.4386
107.9648
139.5380
-5.5814
90.8254
121.3864
-6.1459
110.7567
136.4740
16.8146
75.0704
108.3575
7.2360
109.6951
140.9111
5.7369
111.6490
136.3009
-9.8549
83.8900
110.1130
3.5986
125.8516
152.4980
-11.2948
95.2230
119.6473
9.4649
78.5562
107.2132
-5.4747
66.0913
104.3142
8.2772
124.3079
153.0749
-17.6914
110.8053
144.0061
-17.4957
116.4955
141.6519
-6.4137
98.9005
124.2610
-7.3708
76.6720
103.0297
-1.2608
122.0648
156.1843
-50.5345
129.5729
155.2330
22.9470
94.3129
117.9179
-2.1154
57.7964
92.6869
13.3883
126.5464
152.7744
14.0396


161
It follows that the ith column of A is
dK{0\y)
dui 0
X'iR^iVi Xt$ ZiUi)
- X& ZiUi) G;lUi
X'.Rr'ei
Z\R^ex Gr'i
(5.42)
This expression can be further simplified. Because the matrices Z, R, and G are all
block diagonal, the BLUPs for the ith cluster can be written as
i = (Z'iRr'Zi + Gr^ZR^a,
(5.43)
where c is the vector of composite residuals for the ith cluster. Substituting this
into the bottom portion of (5.42), we find
Z[Rrlei Gr'i = Z'^ia Zii) G^u,
= Z\R-Xc (Z'iRflZi + G~x)ui
= Z'iRr'a (Z'iR^Zi + Gi1)(Z'iRilZi + G~l)~x Z\R~X Ci
= 0.
This implies that only the top p\ rows of A are non-zero. The matrix can be written
compactly as
A =
X'RlD(ei)
0n
J9*9
(5.44)
where D(e) is an n x m block-diagonal matrix whose ith block is e.


118
The next step is construction of (4.54). First, the trace of product of two matrices
Arxc and Bcxr is (Searle 1982)
i=1 j=1
-X
Letting denote the ijth element of h we may write (4.52) as
- V V y)
do'duk
i=i j=i ' *j
p p
*"--EE
dhi^{0,y)
i=l =1
dOjduik
htJ.
Construction of su
follows:
dhi,u(8;y) i
P P
d6jWi lu
Suj = -
-EE*
t=l j=l
dhUli(6]y) i
d0ju>q 1
9/ii,u(9;p)
9hi,u(6;y)
9h2,(9;y)
9h2,u(9;y)
dhPlU)(0,y)
99pui i
dOiun
90pui i
99pu>i
9hUu(0-,y)
5/ii,u(9;p)

d9\w2
ddpLJ2
9hi,u(9;y)
dhiM(6\y)
9h2,u(9;y)
9h2¡u(9;y)
dhpquiQ'llj)
d9\Uq
dOpUlq
99 ICJq
99pUq
99pUlq
=
R\ R%
Rp vec(
-h-1).
vec(h X)
Construction of s7 proceeds similarly.
Corollary 4.3
For the measure (4.51), the slope in direction v is
S = vec(h )'
R' + hJ'
v,
and the direction of maximum absolute slope is
Umax OC m =
-1.
R + J/i vec(/i )
(4.59)
(4.60)


4.6 Local Assessment for Maximum Likelihood Inference 110
4.7 First-order Assessment of the Log Generalized Variance ... 116
4.8 Local Assessment for Quasi-likelihood Inference 125
5 LOCAL ASSESSMENT OF PREDICTIONS 134
5.1 Prediction via First Principles 134
5.2 Predictive Likelihood 136
5.3 Influence Measures for Prediction 138
5.4 Estimation and Prediction in Linear Mixed Models 140
5.5 Local Influence Methods for Linear Mixed Models 146
5.6 One-way Random Effects Model 151
5.7 Residual Variance Perturbation Scheme 154
5.8 Cluster Variance Perturbation Schemes 159
5.9 Application to Corn Crop Data 167
6 MULTI-STAGE ESTIMATIONS AND FUTURE RESEARCH ... 185
6.1 Local Assessment in Multi-stage Estimations 185
6.2 Synopsis and Additional Future Research 186
APPENDICES
A MODEL FITTING INFORMATION FOR SCHOOL DATA .... 188
B MODEL FITTING INFORMATION FOR CORN CROP DATA . 191
REFERENCES 194
BIOGRAPHICAL SKETCH 202
vi


156
Evaluating this at the original model we obtain the ith column of A:
0 0 ... 0 \cxi 0 ... 0
0 0 5 A v*f D 1 111 1 1
i ^ * 2^n
0 0 ... 0 !<7(+i) 0 ... 0
0 0 ... 0 \ani 0 ... 0
Rle
(5.38)
When R = a2I, R'w = <72D(l/u;), and the expression above simplifies, allowing us
to write
A = X-D(e)
and
M = -5
a = 4 (e)[x Z1
XX
xz
-1
X'
zx
ZZ + o2G~l
z'
D(e).
5.7.3 One-way Random Effects Model
Deriving results for the one-way random effects model in more detail should
provide information about the pertubations diagnostic value. First, consider
influence on the intercept parameter estimate, fi. This can be accomplished by
using either m = or m PLD^\u>) as the influence function. The direction of


171
(a) Basic curvatures
(c) ^max for /30
(b) v
(d) ^max for Pc
1.5
1.25
1
0.75
0.5
0.25
(e) ^max for ¡3,
soy
2 4 6 8 10 12
(f) Cmaxs for each of the
Figure 5.2: Corn data; Scheme 1; local influence diagnostics


144
follows. The first step is to maximize the joint likelihood of u and (3 with respect
to /3 for given u. This leads to the first set of mixed model equations (5.17), which
have solution ¡3(y,u) = (X'R~1X)~1X'R~1(y Zu). The next step is to show
that predicting u with its conditional mean given the data and the estimate $(y, u)
leads to the second set of mixed model equations. Starting with the profile predictive
likelihood, we have
L*(u,i3(y,u)-,y)
oc exp ^ ^ (y X0(y, u) Zuj R~l (y X${y, u) Zuj ^u'G^u^j
=exp ( \y'R l (y x$(y,u) Zu)
+ \p\yi u)X'R~l [y X0(y, u) Zuj
+ iu'Z'R~l [y X(3(y, u) Zuj ^u'G~lu)j
=exp ( \y'R_1 (y xp(v,) zuj
+ 1u'Z'R-1 (y X$(y, u) Zuj lu'G"1),
(5.19)


141
linear mixed model are reviewed, and additional tools for assessing predictions are
presented.
5.4.1 Linear Mixed Models
The linear mixed model (LMM) is formulated as follows:
Y = X/3 + Zu + e,
(5.12)
where
f
u
~ N
0
e
\
0
G 0
0 R
5
Y is an n x 1 vector of observations, X is a known nxpj matrix, Z is a known
n x p2 matrix, (3 is a pi x 1 vector of unknown fixed effects, u is a p2 x 1 vector of
unknown random effects and e is a n x 1 vector of unknown residual errors. The
matrices G and R are of order p2 and n respectively.
Littell et al. (1996) provide a host of applications for this rich class of models.
When there are no random effects and R = a21, this model simplifies to the fixed
effects regression model (1.10). The inclusion of random effects and the general
residual variance matrix R permits the modeling of correlated data. This is evident
by looking at the marginal distribution of the data under model (5.12):
Y ~ N{X/3,ZGZ' + R).
(513)
The unknowns in (5.12) are the fixed effects ¡3, the random effects u, and the
elements of the covariance matrices G and R. In this chapter, we will concentrate
on assessing the effects of perturbations on inferences for (3 and u. Point estimates
of ¡3 and point predictions of u can be motivated by three methods: first principles,
mixed model equations, and profile predictive likelihood. See Searle et al. (1992) for
information on estimating G and R when these matrices are unknown.


63
3.3.3 Directed Influence on /3j
Without loss of generality, we can direct influence on /3X:
l^max OC A T
-
(1)
1
-7i
+ (li %u)P
and
a
max /v o 11 i
(T \\rl\
||(1 7i)' + (li ~ *n)P\\ ,
where n is the predicted value of Xu from modeling Xi as a linear combination
of the other regressors. More generally, the maximum local influence on by
perturbing row i is
/
l I
+ (jj xij)0
Ti
Cmax
V
1
-7 j
CT4 T\
IK1 -7 'j)'+ (xij-xij)p\\2.
If the columns of X have all been standardized, then each of the elements of 7 and
of $ is less than 1. This suggests that element i of vmax is likely to be larger than
the other elements. However, the signs of r and (j Xij)0j can work to cancel
each other out. Further interpretation is suppressed in lieu of the more general


201
Zeger, S. L. and Liang, K.-Y. (1986), Longitudinal Data Analysis for Discrete and
Continuous Outcomes, Biometrics, 42, 121-130.
Zhao, Y. and Lee, A. H. (1995), Assessment of Influence in Non-linear
Measurement Error Models, Journal of Applied Statistics, 22, 215-225.


108
Manipulation of the right side of the equality proceeds as:
Iq J'
IQ J'
Iq J'
Iq J'
*1
(/, J'l 0
-
o [J, J'l ...
*
1
h[lQ J' 1
*
It then follows from (4.43) that
P =
h I,
-h /,
Ip [la J'}
IP[lQ J')
*
J
In
-h1 [ Iq J' ]
*
i

Corollary 4.2
= (-a")
i
s
Cr
' i=i

J
p
= + RjJ + JR] + J'hjJ)
j=i


65
i
0.75
0.5
-0.25
-0.5
-0.75
3 4
5
-ll
Figure 3.5: School data; x'18 + u/, vmgx for 0
3.4 Perturbations to the Entire Design Matrix
At this point, the general case of perturbing all nk elements of the design
matrix is considered. We use W to denote the perturbations in matrix form:
a>2+i . Wfen-n+l
... ... kJfcnn-f 2
... ... ^fenn+3
W3n Cdkn
3.4.1 Building Blocks
The derivations in this section only require the presence of an intercept in
the model (or centering of the response y). However, for ease of interpretation, we
assume the columns of X have been centered and scaled. The perturbed likelihood
is
w =
Wl
^n+1
U)2
^n+2
UJ3
w
W2n
and u>j to denote the jth column of W.
LJ; y) (3.18)


89
on obtaining optimal estimating functions (OEFs) (Godambe 1976; Crowder 1986;
Crowder 1987; Godambe and Heyde 1987; Firth 1987; Godambe and Thompson
1989; Godambe 1991). To this end, the efficiency matrix of a set of p estimating
functions is defined as
ES(h) = (E
'dh''
~de
)-IV [ft] (E
(4.2)
Functions that minimize this non-negative defnate matrix are considered optimal.
Roughly speaking, this index of efficiency is small when the variance of the
estimating functions is small and the average gradient is large. Provided that the
estimating equations arise from maximizing or minimizing some function, we may
interpret that large gradients indicate the function is not flat around its extremum.
Assuming that a fully parameteric form for the data is assumed, the score
functions are the unique OEFs up to a scalar multiple (Godambe 1960; Bhapkar
1972). However, with fewer assumptions about the data, it is generally not possible
to find a unique set of optimal estimating functions. Instead, optimality is bestowed
upon estimating functions within a certain class. One such class of estimating
functions are those that are linear functions of the data.
Under mild regularity conditions, the asymptotic variance of 0 is in fact given
by (4.2) (Morton 1981; McCullagh 1983; Crowder 1986). Thus, minimizing Eff(/i)
is equivalent to minimizing the asymptotic variance of the parameter estimates.
Note that the first interpretation of the efficiency matrix is appealing in that it is a
finite sample index, but somewhat awkward in that it is based on a property of the
estimating functions. The second interpretation is appealing in that it is based on a
property of the estimates, but limited in that it requires asymptotics.
For the purposes of applying local influence diagnostics we assume only that
the estimating functions have been standardized. That is, given a set of unbiased
estimating functions, h*(0; y), we utilize the standardized estimating functions given


180
(e) v3
(f) |u3| star-plot
Figure 5.7: Corn data; Scheme 2; local influence on


54
where 7*1 is the regression parameter estimate corresponding to Xi when predicting
Xk with the other columns of the X matrix. The maximum curvature is given by
2
2K||2
%\r' + j3xr'k 7*i r + Pirk
- h (*
2,-2
+ 7*r
*
(3.11)
When the partial correlation between Xx and X* is zero, CmSiX achieves its minimum
value of -?2, yielding vmax oc r*. In the event that X* is a linear combination of the
other regressors with non-zero 7*1, Cmax = 00 and vmax oc r. Between these two
externes, the direction of maximum curvature is a weighted average of the residuals
and the adjusted kth regressor. These quantites are used to detect outliers and
measure leverage, respectively. If the partial correlation between Xi and X* is as
large as that between Xi and y, then the two vectors are equally weighted. Note
that this weighting is invariant to scale changes in the columns of the design matrix
as well as the response.
3.2.4.1 Directed influence on 0
Without loss of generality, we again consider perturbing the first column.
The rank-A; acceleration matrix is
F "P
Tj) +MH Crr+r (3.12)
MV ~ llril
The first eigenvalue of this matrix is unique, and yields a maximum curvature and
corresponding direction of
c 2
'-'max -7
2+ ITU2
UrTU*
and
Vi cx r PiVi.


105
the observed information matrix. The matrix A is also easily computed provided
that the expressions for the perturbed estimating functions are known. This
computation was demonstrated many times in Chapters 1-3 for maximum likelihood
Computation of A could be difficult when the estimating equations themselves are
not closed-form. An example would be ML inference for generalized linear mixed
models (McCulloch 1997).
4.5.1 Computation of P
In this section, a formula for P is given so that it, like J, it can be computed
without re-estimating the model. To begin, we present the following lemma, which
provides an identity used to construct P.
Lemma 4.2
Let Ou be the solutions to the p perturbed estimating equations hu{0\ y) = 0.
-hi Iq P
Iq J'
J
i = 1
(4.36)
where
hi = the ith column of h
Qi Ri
R[ hi
d2hicJ{0; y)
dwdu)'
d2hiuJ(0:y)
dudO'
d2hlu]{6\ y)
{ dodo'
J = -h 1 A.


80
Table 3.3: Hills data; curvatures using X + W perturbation. Model I contains
DISTANCE and CLIMB. Model II contains DISTANCE and CLIMB-RATE
Estimates in LD(u>)
or LRW()
Model I
Model II
d2
Xi = 30.47
Ai = 28.52
3
Ai=21.15
Ai=16.41
A2=16.49
A2=16.25
influence when fitting the regression model for winning time as a linear combination
of DISTANCE and CLIMB. However, as was evident in Figure 1.5(c), the two
regressors are linearly related. An alternative model can be fit by replacing climb in
feet with climb in feet per distance travelled, CLIMB-RATE This regressor does
not have the strong positive association with distance that the original coding of
CLIMB has. This should reduce the collinearity in the model, thereby reducing
sensitivity to perturbations in the X matrix.
Using standardized regressors, the new model is fit. Column 2 of Table 3.3
shows the curvature values for this new model. Cmax for directed influence on ¡3
has been substantially reduced, indicating that the MLEs for the second model are
indeed less sensitive to perturbations in the X matrix.
Figures 3.9(a), (b) and (c) provide the profile plots of the absolute value
of the directional vectors that accompany the curvatures in Table 3.3, Column II.
All three plots tend to indicate DISTANCE values as being more influential than
CLIMB-RATE values, and all three plots show substantial change from those in
Figure 3.8. The plot for d2 now indicates the DISTANCE value for observations 7
and 18 as influential, rather than just 18. No elements stand out in t>max, while case
lls DISTANCE value is a very substantial contributer to u2.


157
maximum local influence for both measures is given by
"Omax OC ft d\
oc D(e) [X Z]
o-j?+ crZ+(T2/ng
oc £)(e)
(! al+a2/m )^ni
OC
g2/"l Cl
<72+ 1
3
1
oc D(e)
In
.
1
M
3
1
<72+ct2 /ng .
(5.39)
o-2/ng
a2+a2]ng e9
where e denotes the residuals for cluster i. Thus, the direction of largest local
influence is proportional to the residuals, but with each cluster scaled according to
its size. The coefficients upweight observations in smaller clusters and downweight
observations in larger clusters. The reason for this is that p is not a simple average
of the observations: the numerator of p in (5.32) can be written as
9 Tli
Vij
^ ^ ^ n.(r2 l fj2
=1 i=i n'ac +
This indicates that observations from smaller clusters receive more weight. Note
that if either the design is balanced or of = 0, then vmax
oc e.


145
since X'R x(y X0(y,u) Zu) = 0. Examining the exponent, we find
~\[y'R~\y XP{y, ) Zu) u'Z'R-\y X0(y, u) Zu) + u'C^u^j
= -\ [y'R-ly ~ y'R-'XPiy, u) 2u'Z'R-ly + u'Z'R-xXp(y, u)
+ u'ZlR-lZu + u'G1u
c-i(-y'R~xX (X1 R~x X)~x X'R~x (y Zu)
- 2u'Z'R~1y + u' Z'R~x X (X'R~x X)~l X' R~l [y Zu)
+ u'Z'ir1Zu + u,G~1u
oc -- ^ 2u'{Z'R~x -
+ + Z'RXZ + G~x)u^
= -i ^'(Z'iJ H-1X(X,i21X)-1X,)H1Z + G-1)u
- 2u'(Z'(I R-1X{X'RT1X)-1X'))RrlySj
= ~ (u'{Z'P'R-lZ + G_1)u 2u'Z'P'R-'y^j,
where P' = I R~x X {X'R~l X)~l X'. It follows that
/(u|y, 0(y, u)) oc exp ^ ^ (iu {Z'P'R xZ + G-X)u 2u'Z'P'R-xy) j
This is proportional to the kernel of a normal distribution with a mean of
(5.20)
(5.21)
(z'p'r-'z + G-'y'z'p'n y
1 \ 1 rj< r>' D1.
(5.22)
and this predictive mean is the point prediction for u.


no
Hence, another form of the acceleration is given by
r
+ J'[X>7,G(] J
M =
I J'K'
S
L -J
I
KJ
*=1
i=1
Iq J'
1 Q
J
(4.46)
-i
Note that in all computations, J = h A as per Theorem 4.3.
4.6 Local Assessment for Maximum Likelihood Inference
In this section some brief examples when estimation is performed by ML are
given. The influence measures addressed include linear combinations of parameter
estimates, ratios of parameter estimates, the likelihood displacement, the profile
likelihood displacement, and the Kullback-Leibler divergence. Emphasis is placed on
constructing the building blocks of local influence. From Theorem 4.3, J = L 1 A.
The individual measures determine and the remaining building blocks:
dra(u;,7(0))
K =
duj
si =
dm( a;,7(0w))
dei
S =
dj(Oui)
82m(a;,7(g)) d2m(u,i(8)) i
duidcj' I dajdy(8)' I
dim(uj,"i(8)) i 32m(w,7(9)) i
d'y{8)du' 1 d~t(6)fry(e)' 1
4.6.1 Linear Combinations
Consider a linear combination of the parameter estimates under perturbation:
m = x'Ou.
The building blocks of first-order local assessment are:
7(0W) = x'Qu
3^-0
K = x'
s7 =
0
1.


30
(2.9):
^rriax OC
a
Xx X{1)
r i
= iiTT [*. -X'lDT'i]
<7||i*l
1
llnl
rri
1
-7i
oc ri
and
= 2||
<7 *T
rrill = To-
Therefore, the direction of maximum curvature for directed influence on is
proportional to the jth adjusted predictor.
2.4.3.1 Directed influence on 0
The acceleration matrix for the profile likelihood displacement for ¡3 is the
first term of (2.10):
(2.12)
This matrix has a single non-zero eigenvalue of with multiplicity k, implying that
there are an infinite number of directional vectors that have maximum curvature.
Although not discussed by Schwarzmann (1991), one set of orthogonal solutions can
be motivated by considering the canonical form of the model:
Y Zol T c,
(2.13)


15
variance. Thus, case-deletion can be thought of as a rather drastic perturbation:
one that results in a variance > oo for the ith observation. Case-deletion is referred
to as a global influence technique because it assesses the effect of perturbations that
are not local to the null perturbation in 2.
1.4.2.1 Influence on the full vector of parameter estimates
Figure 1.10(a) shows the direction of maximum curvature for the Scottish
hills data set. Again, element 18 dominates the direction of maximum curvature.
This implies that the precision of observation 18 is the most influential on the full
set of parameter estimates.
1.4.2.2 Influence on a2
Using the profile likelihood displacement, influence directed upon a1 alone
can be assessed. Observation 18 is again implicated by umax as being most locally
influential (Figure 1.10(b)).
1.4.2.3 Influence on individual regression parameter estimates
Figure 1.10(d) shows vmax when isolating effects on /^ist- Although there are
several points that stand out, observation 7 is identified as being most influential
since element 7 is large. This race is also implicated as being locally influential in
the plot of umax for /3ciimb (Figure 1.10(c)). In fact, the DFBETAS diagnostic
(Belsley et al. 1980) also identifies race 7 as being influential on Recall that
the value of DFBET AS^j measures how the jth parameter estimate is affected
by deleting the ith observation. Writing the n DFBET AS for the jth parameter
estimate in vector form, it is shown in Section 2.5 that this vector is proportional to
the following Hadamard (element-wise) product of vectors:


84
Figure 3.10: School data; X + W pert.; vmax for directed influence on /3
Figure 3.11: School data; X+W pert.; Column profiles of |umax| for directed influence
on /3. Thick solid line- SALARY, thin solid line- WCOLR, dot-dashed line- SES,
small-dashed line- TSCORE, large-dashed line- MOMED.


81
Figure 3.7: Hills data X + W pert.; rmax for a
^.2
Example 2
We now consider applying the X + W perturbation to the school data and
examining directed influence on ¡3. The purpose of this example is to demonstrate
alternative methods for displaying t>max when the number of regressors and/or
observations is large.
Figure 3.10 is the plot of umax for directed influence on ¡3. Elements 1-20
correspond to SALARY, elements 21-40 correspond to WCOLR, elements 41-60
correspond to SES, elements 61-80 correspond to TSCORE, and elements 81-100
correspond to MOMED. The two regressors that have little association with other
explanatory variables, SALARY and TSCORE, have small elements. In contrast,
some of the elements for the other regressors are more substantial. This plot does a
fair job of highlighting regressors, but a poor job of highlighting observations.
Figure 3.11 is the profile plot of |i>max|. If is easy to see that cases 3 and 18
have multiple regressor values that are influential. In addition, one of the values for
case 2 is influential. However, is to difficult to keep track of which line represents
which column. Another option is to plot profiles of each row, as in Figure 3.12.
Here, some of the elements of observations 2, 3, and 18 are highlighted.


13
i
0.75
0.5
-0.25
-0.5
-0.75 (
-1
Figure 1.8: Hills data; alternate rendering of t>max
signs of the elements are suppressed: a star plot. Here, each element is displayed
as a point, with the distance from the origin equal to its absolute value. The q
elements are simply displayed around the point (0,0), starting in the first quadrant
and ending in the fourth quadrant. The plot is completed by labelling the points
with indices. Labeling all points can result in cluttered plot (1.9(a)), and so a better
option is to only label the largest elements (1.9(b)). In these renderings, element
18 is again noticeably large, but our attention is also drawn to element 7. This
observation has the characteristics of an influential observation: it has the highest
climb and one of the longest distances (Figure 1.5(c)), as well as the second largest
residual.
Unlike the index plot, the star plot does not give the illusion that the
indices are in a specific order. However, the suppression of element signs can be an
advantage or a disadvantage. The advantage is that when looking at several plots
of directional vectors, suppressing the signs helps the analyst identify patterns by
emphasizing element size. The disadvantage is that the diagnostic interpretation of
the signs is lost.


102
Then,
d2m(u,y{GlJJ)) __ d
'dm(S) du:
da2 da
du:'
da
d
'dm(d)
da
du:'
d
dm(S)
da
du'
dm(6) dv
du:' da
du:' d2m(j(Gw))
da du)d(jj
,d2m(S)
v ^v.
v + 0
dudu'
, ( d \dm(l
~v VaJ
dm(S) dS
duS
v
= V
V
d_
dui
dm(S)
dS'
v.
q+r
dS + g dm{6) d26i
=1
du:
dSi dudu'
dd' d2m(S) dd y, dm(S) d2Sj
du dd' du:' + 4-f dSi dudu'
v.
v.
i=l
r
d'd2m(S) dS ^ dm(5) d2'yi(GlJj)
+ Iu>
(4.29)
du: dddS1 du:' y-f d~/i(Gu.) dudu'
where the last simplification occurs because d2u)i/dudu' 0. Examining the first
term of d2m(S)/dudu', and evaluating at the null perturbation we find
dS' d2m(S) dS
du: dSdd1 du:1
dui' 07 ( d2m(6)
diii
dhi'
0
00/ dill
0
dSdS1
0
diii'
Iq J'K'
d2m(w,7(0)) i
oLdJ lo
a2m( dud, (0)' 10
d2m(uM9)) I 02m(oj,7(fl)) I
07(0)0w' 1 07(0)07 (ey 1
Iq
KJ
(4.30)


95
If m(0) is a local minimum, maximum or saddlepoint of the influence graph,
then the gradient is zero. Here, a first-order local assessment does not provide any
diagnostic information. Instead, we can turn to a second-order approximation of the
influence graph.
4.2.2 Second-order Local Assessment
A second-order Taylor series expansion of m is given by
/ \ ,x dm, 1 d2m,
m(a) m(0) + (a 0) + --^-(a
dm
(0) + ag^v + ~ov
a2 dm?
2 dudu
v
= ra(0) + arn'v + v'Mv,
z
(4.16)
where the derivatives are again evaluated at a 0. The quantity A = v'Mv also
has a geometric interpretation: it is the rate of change in the slope of the lifted line
associated with direction v, i.e. it is the acceleration of the curve. The matrix M
is the acceleration matrix, and the eigenvector associated with its largest absolute
eigenvector is the direction of maximum absolute acceleration. If the gradient
vanishes, the second-order approximation simplifies to:
a2 ..
m(a) m( 0) + v'Mv. (4-17)
z
Both the acceleration and the acceleration matrix have diagnostic value,
particulary when the gradient vanishes. In this case, A can be interpreted as the
approximate effect of a size-\/2 perturbation in direction v. In addition, suppose
that we have perturbations in two different directions: and u6. If \A\ is k
times larger than \Aq\, then perturbation u>s must be \fk times larger than to
have the same effect. Finally, the eigensystem of the acceleration matrix can be
used to identify influential perturbations. The largest absolute eigenvector of M
is the maximum approximate effect of a size-v^ perturbation, and the associated


90
by
(VI**])-1**. (4.3)
This standardization has three implications. First, V(/i(0;y))_1 is equivalent
to (4.2) and hence equivalent to the asymptotic variance of the parameter estimates.
In addition, the standardized estimating functions are optimal in the following sense.
Given a set of unbiased estimating functions, |A*(0;y)}, the OEFs amongst the
class of linear combinations of the h* are given by (4.3). Hence, the standardized
estimating functions are an optimal rescaling of the original estimating functions.
Note that the negative sign is present in (4.3) so that the score equations, rather
than the negative of the score equation, are considered standardized. Finally,
standardized estimating functions are information unbiased (Lindsay 1982),
meaning that the following equality holds:
= E[hh'\ = V[/i]. (4.4)
Hence, V[0] = E[dh/dO']. This fact is used later in the chapter to derive influence
diagnostics for confidence ellipsoids.
4.1.2 Perturbations
Perturbations are again introduced as a vector of q real numbers, u>. These
perturbations may affect the data, the model and its assumptions, or perhaps even
just the estimation technique itself. Examples of data perturbations are minor
changes or rounding errors in the response and the explanatory variables. An
example of model perturbation is the variance perturbation scheme used in Chapters
1 and 2. Finally, perturbing the contribution of each observation to the estimating
functions by case-weights is an example of perturbing the estimation technique.
This perturbation does not affect the distributional assumptions of the data, but
rather how each observation is utilized for inference. Another perturbation to the
-E
dh
W
h{0;y) = -E
dh*'
~de


68
This result implies maximum local influence on a2 is achieved by perturbations in
each column that are proportional to the residuals scaled by the corresponding /%.
3.4.3 Directed Influence on /3
In this section, we derive the full eigensystem of F where
FW = ^ (ik r ~ P <8> x) ((X'X)-1 (8) r' p' (X'X)-1X') .
The following theorem provides the full eigensystem of F for the X + W
perturbation in a regression analysis. Note that eigenvalues in the theorem appeared
in Cook (1986), but the eigenvectors appear here for the first time.
Theorem 3.1
Recall that (fj is the jth eigenvector of X'X and that Xtpj is the jth principal
component. Letting Sj denote the jth eigenvalue of X'X, the k non-zero eigenvalues
and eigenvectors of F^ are
Aj = 2(-^+ M-) vjoc k-j+l G
for j = 1,..., k.
Proof of Theorem 3.1
Because the algebraic expressions are so long, the eigensolutions are confirmed in
two steps. Noting that g-- and

32
Y ~ N(X/3,a2D(l/u:)), where D(l/u>) denotes a diagonal matrix with ith element
A. The null perturbation is u)0 = 1. The n basic perturbations would be the local
versions of the n Cooks distances (Cook 1977; Cook 1979). In addition, the n basic
perturbations applied to LD^j\uj) are local influence analogs of the n DFBETAS
for /3j.
A second interpretation of this perturbation scheme comes from the idea
of case-weights. Case-weights are used in an analysis when each observation has
been sampled with unequal probability. Under disproportionate sampling, cases are
weighted by the inverse of their sampling probability in order to obtain unbiased
regression parameter estimates. In the normal linear model, this weighting is
numerically equivalent to perturbing the error variances. Hence, this perturbation
scheme can be used to examine sensitivity to the assumption of random sampling.
2.5.1 Building Blocks
To begin, we develop the perturbed likelihood, which is
77 1
L(0: y) oc -- logo-2 (y X(3)'D(u;)(y X/3)
= logo1 ^y'D(w)y.
The next step is construction of A. Partial derivatives with respect to (3 are as
follows:
9Lg%V) = -(-2 X'D(u)y + 2X'D(U)XP)
= X'D(u)y,
and the partial derivative with respect to a2 is
dLu(0\ y)
da2
5 +y'


176
do. Each element is displayed as a point, with the distance from the origin equal to
its absolute value. The plot is completed by labelling the points with indices and
drawing rings to represent the benchmarks and
Figure 5.5 contains index plots and a star plots of the three distinct
eigenvectors resulting from the cluster perturbation. Element 12 of umax is almost
three times the value of equally contributing elements (plots (a) and (b)). Element
11 of v2 is nearly two times that of e.c.e. (plots (c) and (d)), while element 8 of v3
is about two times that of e.c.e. (plots (e) and (f)). Nonetheless, the second two
eigenvectors tend to be more noise than signal.
To explore further how cluster 12 is affecting inference, Figure 5.6 gives
directions of maximum curvature for directed influence on each of the individual
BLUEs under the original parameterization. Element 12, together with element 11,
appears prominantly in umax for both of the slope estimates. This indicates that the
slopes, rather than the intercept, are being influenced by county 12.
To assess effects of the perturbation on the set of i, Figure 5.7 gives the
three eigenvectors of M The direction of largest curvature shows that element
11 is more than two times that of equally contributing elements, while element 12
is roughly three times that of e.c.e. in the second eigenvector. Finally, element 8
is most prominent in the third eigenvector. Because element 12 stands out in v2,
but not umax, these plots are inconclusive as to whether county 12 is influential as a
whole on the BLUPs.
Finally, we address the main question of interest- whether or not the
predicted mean hectares per segment are sensitive to the perturbation. Since there
is only one prediction per county, we can use the perturbation scheme to produce
cluster-based analogs of BasicFITS and MaxFITS (Figure 5.8). The MaxFITS values
show that all of the predicted means show comparable sensitivity to perturbation.


140
The influence measure following way. First, define the following modified predictive likelihood displacement:
PLD*(u) = 2[PL(u- y, 0) PL(: y, 0U)\. (5.10)
This differs from PLD(w) in that the prediction for u is not perturbed. Also,
assuming y and u share the same density and that all elements of y and u are
independent, we have
m m
PLD*(u>) = 2[log/(y|/3) + ^log/(|/3) log/(y|0J 5^1og/(fi<|3j]
1=1 1=1
m
= 2[log/(y|/9) log/(y|^w)] + 2 ^(log/(|/3) log f(i\0u)) (5.11)
=1
m
= LD(w) + 2^dJ(w),
*=i
where d\ (a;) is the deviance residual of perturbation for the ith prediction. Note that
the prediction for u is the same in both terms of d*(u). Technically then, d*(cj) is a
measure of the pertubations effects on the estimated conditional mean of u rather
than a change in the prediction for u.
We note that under the same assumptions given above, the predictive
likelihood displacement (5.7) can be written as
m
PLD(u) = LD{u) + 2^2
=i
where
di(w) = log f(i¡/3) logf(itU¡0J.
5.4 Estimation and Prediction in Linear Mixed Models
In this section, the linear mixed model is presented with point estimates and
point predictions motivated in three ways. Then, local influence techniques for the


155
additive fashion. The perturbation changes the variances but leaves the correlations
among the error terms unchanged. The resulting local influence diagnostics can
detect influential observations without upsetting the relationships assumed to exist
among the residual errors.
It simplifies to the error variance perturbation given for regression when
R = a21.
5.7.2 Computational Building Blocks
To derive A = dhu(0; y)/du>'\o, define e = y Xfi Zii, and X* = [X Z].
We begin with the perturbed joint likelihood. Ignoring terms that do not depend on
both 6 and u: we have
£*(, v) = ig(f(y I u)g(u I £))
* -\[(y X-0)' RZl (y X-e)\
= -\ [v'R-Jv 2ffx-'R-Jv + e'x-'R-'x-e].
Taking derivatives with respect to 9, we find
fa. dLu(u,0;y)
K{0,V)= gg
= -i [~2X"R-ly + 2X"R-'X$]
= X"RZ\y- X-B).
Second partial derivatives are given by
^K(6-y) = -x-'Rr^K\v-x-).


35
where r is the ith residual, Cj is the jth column of X(X'X)~l, Cij is the ith element
of Cj, and s2 = is the mean squared-error estimate of a1. The vector Cj can be
written in terms of the jth adjusted regressor by partitioning (X'X)-1 as per (2.7).
For j = 1,
Cl
X, X(1)
1
i
-7i
Using this equality, the vector of n DFBETAS for is proportional to
/ i \
l-hll
1
I-/122
Tj-
(2.19)
1
\ 1hnn /
Since umax oc r rj, observations with large leverage stand out more in a plot of the
n DFBET AS for ¡3j than in a plot of vmax- This algebraic relationship between
Umax and DFBET AS has not been noted by previous authors.
2.5.4
Directed influence on (3
The acceleration matrix for directed influence on $ is the first term of (2.15):
FW = y2 [D(t)HD(r)].
The eigensystem of the matrix does not have a closed-form. However, some idea of
the directed influence of this perturbation on $ can be ascertained by utilizing basic
perturbations. The ith basic perturbation corresponds to modifying the variance of
r/3i
only the ith case, and its curvature is the ith diagonal element of F (Cook 1986):
Chi = Tiuhn.
o
(2.20)


10
analyst to determine which observations would have the most effect on parameter
estimates in the event that there is measurement or rounding error.
1.4.1.1 Influence on the full vector of parameter estimates
Atkinson (1986) analyzed a data set of record times for 35 different hill races,
as recorded by the Scottish Hill Runners Association (Table 1.1). Scatter plots
indicate the positive association of the three variables: winning time, race distance
and race climb. (Figure 1.5). The influence of perturbations on the parameter
estimates from a regression of winning time on DISTANCE and CLIMB is of
interest. For convenience, the regressors and response have been centered and scaled,
thereby eliminating the need for an intercept. Thus, the full vector of p parameters
S 0 = (/?dist > /^clirab ? )
Figure 1.6 shows umax plotted against the indices 1 through q = n = 35 for
the Scottish hills data. Element 18, which corresponds to perturbation of the 18th
observation, is the dominant contributer to the direction of maximum curvature.
This observation is from a short race with little CLIMB, and yet it has a large
record time. Similar races have much smaller record times, indicating that race 18
is an outlier. Atkinson (1986) has actually suggested that the winning time for this
race may have been 18.67 minutes rather than 1 hour and 18.67 minutes.
Interestingly, it can be shown that the direction of maximum curvature for
Scheme 1 is proportional to the residuals. Thus, if we standardized the vector
of residuals to length 1 and plotted them against case indices, we would obtain
Figure 1.6. This implies that parameter estimates are most sensitive to a set of
small additive measurement errors when those errors tend to exaggerate (or correct)
the models error in estimating each data point, with the most exaggeration (or
correction) occurring for the most poorly predicted points.


142
5.4.2 Inference by First Principles
The best linear unbiased estimates (BLUEs) of /3 are the generalized least
squares estimates:
¡3 = (X'V-xXylX'V-ly, (5.14)
where V = ZGZ' + R. Here, best refers to minimum mean-squared error of
estimation. The estimates are equivalent to the MLEs derived from the marginal
distribution of Y.
The best linear unbiased predictors (BLUPs) for u are given by
u = GZ'V'c, (5.15)
where c = y X¡3 is the vector of generalized least squares or composite residuals,
the latter term being coined by Hilden-Minton (1995). Here, best refers to
minimum MSEP.
Note that the BLUEs and BLUPs do not require normality- only the model
and covariance structure specified in (5.12). Detailed derivations can be found in
Searle (1971) and Searle et al. (1992).
5.4.3 Inference by the Mixed Model Equations
Henderson et al. (1959) derived estimates for the LMM by maximizing the
joint log-likelihood of u and /3:
L{u, /3; y) = log(/(y | (3, u)g(u \ (3))
(5.16)
oc -- [(y X/3 Zu)' R1 (y-X(3 Zu) + u'G 'u] .
Zi
This likelihood is also known as an h-likelihood (Lee and Nelder 1996). The solutions
that maximimize (5.16) are in fact the same as the BLUEs and BLUPs. The
resulting estimating equations are called the mixed model equations (MMEs) and


151
where /i22 comes from the partition
h =
hn hi2
/l21 ^22
When influence on a linear combination, x'$ + z', is of interest, the results of
Section 2.7.1 can be used. Letting x*' = (x',z'), it follows from the argument of
Theorem 2.1 that
Cmax
*\\A'h ^l2
i^11 11
-i
Vr,
oc A'h x*,
where ki
-1/2 *
h x
(5.30)
5.6 One-way Random Effects Model
The one-way random effects model can be used to provide insight into the
diagnostic value of locally assessing the linear mixed model. Some preliminary
results for this special case of the LMM are presented here, so as not to clutter later
explanations.
The (unbalanced) one-way random effects model is
Vij = n + Ui + ij i = l,...g j- 1,... m,
(5.31)
with V(u) = a2 and V(e) = a2. It is a special case of the variance components
model (5.24) with a = 1, X = ln, and /3 = n, and is also a special case of the
two-level model given in (5.25). If all n, are equal to n/g, the design is said to be
balanced. It can be shown that
Vs &
^i=1 0-f+0-2/n;
/*= 1
2^i=l (t2 4-/t2 I
and i
ac + crVni
(5.32)
<72+ct 2/m
where and c are the average response and average composite residual, respectively,
for the ith cluster (Searle et al. 1992).


70
Finally, for j ^ j*, the orthogonality of the solutions follows:
(fPjr-P XtpjYitpj, r ¡3 X = r'r (fijX'r ip'jP r'XlPj* + $ P
= 0 0 0 + p'p
= 0.
3.4.3.1 Interpreting Theorem 3.1
The closed-form eigesolution in Theorem 3.1 gives insight into how slight
changes to the observed explanatory variables affect inference. To interpret the
theorem, consider maximum curvature, as per (3.25):
C
= 2(f+).
Ok
(3.29)
The eigenvalues of X'X are traditionally used to assess the amount of collinearity
in the regressors. If 6k is small, there is redundancy in the explanatory variables,
and this causes the inversion of X'X to be unstable. If this inversion is unstable,
so too are the regression parameter estimates, $ = (X1 X)~lX'y. The first term in
Cmax is large when 6k is small, giving further evidence that the estimates are more
sensitive when collinearity is present. The second term of Cmax is also interesting.
It can be considered a naive measure of goodness of fit; when the relationships
between the explanatory variables and y are strong, ||/3||2 is large and a2 is small.
Hence, better fitting models are more sensitive to perturbation. However, ||/3||2/(72
typically grows whenever additional variables are added to the model, indicating
more complicated models are also more sensitive to pertubation. In fact, ||/3||2/<72
bears some similarity to R2:
ss \\xpf ||X0||2
2 ^regr _
Rz =
SS
y'y n(a2 + ||X/3||2)


174
Of course, the main goal of the analysis is to obtain predictions for the
mean hectares per segment in each of the 12 counties. To determine whether these
predictions are sensitive to Scheme 1, a variation of BasicFITS and MaxFITS is
used. First, the 12 Cmax values for directed influence on county mean predictions
are obtained. These are used to assess the relative sensitivity of the predictions.
Next, the 37 basic curvatures, Cb, are calculated for individual segments as directed
upon the prediction for the corresponding county. This allows us to examine how
much one-at-a-time perturbations within a cluster affect the prediction of that
clusters predicted mean. Finally, the 12 values and the 37 Cb values are
plotted together (Figure 5.4(b)). Here the Cb values are plotted against case indices,
and the Cmax values are plotted against the case indices of observations in the
corresponding county. (Plot (b) is essentially a BasicFITS/MaxFITS plot when only
one prediction is of interest in each cluster.) The implications for the corn analysis
are two-fold. First, the prediction for county 12 is clearly the most sensitive, and
this is due largely to the effects of perturbing case 33. Second, the influence that
case 33 showed on ¡3soy, i, and 7 (Figures 5.2e, 5.3a, 5.3g) does not translate into
noticeable influence on the overall predicted means in the other 11 counties.
5.9.4 Scheme 2: Cluster-based Variance Perturbation
Having seen the influence of perturbing case 33s variance, we may want to
assess the effects perturbing the variance associated with county 12. To this end, we
implement the cluster-based pertubation scheme suggested in Section 5.8.3. Since
there are only 12 perturbations, we implement the strategy mentioned in Section
2.6 including the benchmarks -^= = .577 and -^ = .866 as horizontal
lines. This should prevent us from confusing somewhat large elements with truely
prominent ones. Also, directional vectors are rendered as star plots as well as index
plots. Recall that a star plot is merely an alternative representation that suppresses
the signs of the elements, but does not give the illusion of ordering that index plots


147
Beckman et al. (1987) assumed the fixed effects as well as the variance parameters,
(cr2, er2,..., er2), are jointly estimated by ML. Two perturbation schemes were
introduced. The first scheme perturbs the ith residual variance to u;a2, while the
second scheme introduces a perturbation to the variance of each random effect: the
variance of the jth element of Wj becomes u^-cr2. While both of these perturbations
technically perturb the model assumptions, the major goal is to assess the influence
of individual observations and clusters, respectively.
Lesaffre and Verbeke (1998) emphasized applications to longitudinal data.
They considered the following linear mixed model:
y¡ = Xi/3 + ZiUi + 6j (5.25)
where
yi is vector of observations,
Xi is an n x p matrix ,
Zi is an rii x t matrix,
Ui is a vector of t random effects and
Si is vector of n errors.
It is assumed that V(ui) = G, V(e) = Jnj, and that yi and yv are independent for
i / i'. The model permits the inclusion of random slopes, and the matrix G can be
as general as needed. Also note that the model has a total of p2 = gt random effects.
In the context considered by Lesaffre and Verbeke (1998), the subscript i
corresponds to the ith subject, from whom repeated measurements are taken over
time. The model implies that the log-likelihood is the sum of contributions from each
of the subjects, and by allowing each of the m weights to vary from 1, a subject-level


75
Now we can plot /51)W for some selected values of and $2, considering values
of a ranging from, say, y/2 to V2. This eliminates perturbation matrices with
Froebenius norms that are larger than that of the unperturbed X matrix. Figure 3.6
shows plots of fii'U for four sets of parameter estimates. The solid lines represents the
parameter estimate in the direction i>max for the two-column perturbation, while the
dashed line represents parameter estimates in the direction umax for the one-column
perturbation. All of the influence curves in plots (a) and (b) have a value of .2 under
the null perturbation, i.e. when a = 0. Likewise, the curves in plots (c) and (d) are
all equal to .5 at the null perturbation. In all plots we see that the single column
perturbation does not have the same effect as the dual column perturbation. The
discrepancy between their effects is larger when /?2 is larger (plots (b) and (d)).
3.4.4 Influence in Principal Components Regression
Consider the canonical form of the model, as given in Chapter 1:
Y = Za + e, (3.35)
where Z = Xip and a = (p'/3. The following theorem proves that the direction of
maximum curvature is the direction of maximumum directed influnce on 0*.
Theorem 3.2
Let a. denote the MLEs of a for model (3.35). The pairs \j, Vj given in Theorem
3.1 are the maximum curvature and direction of maximum curvature for directed
inuence on j = 1,..., k.


22
The q x q matrix F = 2A'(L *)A is the acceleration matrix. It is easily
obtained because A is a function of Lu(6;y), and L is typically computed
as part of ML estimation since it serves as an estimated asymptotic covariance
matrix for the parameter estimates. Hence, computing the curvature in a particular
direction is straightforward. In addition, the diagonal elements of the acceleration
matrix are the q basic curvatures. Finally, it is well known how to obtain the
maximum value of a quadratic form such as (2.1) (Johnson and Wichern 1992,
pp. 66-68). Specifically, Cmax is the largest eigenvalue of F, and vmax is the
corresponding eigenvector. Hence, finding Cmax and umax is numerically feasible.
The acceleration matrix can provide additional information. For example, the
second largest eigenvalue is the largest curvature in a direction that is orthogonal
to max' Hence, its corresponding eigenvector indicates a direction of large local
influence that is unrelated to the first one. Additional elements of the spectral
decomposition of F can be used in a similar fashion.
A consequence of this derivation is that it shows how local influence
techniques reduce the dimensionality of examining LD(uj) via a small number of
curvatures and directional vectors. It is well known that the number of non-zero
eigenvalues of a symmetric matrix such as F is equal to its rank. Since it is a
product of other matrices, the rank of F is no more than the minimum of their
ranks, which is typically p.
In the regression examples given previously, p = 3 and q = 35, meaning that
the local influence approach can summarize the 35-dimensional surface LD(oa) with
three curvatures and directional vectors. Similarly, LD^(u>) was summarized with
just two, as shown in Figures 1.12(c) and (d).
Despite the advantages of the eigen-analysis of F, one difficulty that arises
is the occurence of eigenvalues with multiplicities greater than one. For example,
if Cmax = Ai has multiplicity greater than 1, then there is no unique umax to plot.


150
et al. 1992)
where
h11 h12
h21 h22
-/i11 = (x'y-'x)-1
-h21 = -Gz'r'i/i11
-/i12 = _/i21'
-/i22 = (Z'R XZ + G-1)-1 h21X'VlZG
(5.28)
Once a perturbation is chosen and the perturbed estimating functions are
constructed, computation of the acceleration matrix (5.26) is straightforward.
5.5.3 Assessment for Subsets of Point Inferences
Influence measures for subsets of the perturbed point inferences can also be
developed. If only the fixed effects are of interest, then assessment can be done
using the marginal model (Beckman et al. 1987; Lesaffre and Verbeke 1998). More
generally, a profile predictive likelihood displacement can be defined to assess the
a/ /S /
change in a subset of fixed and random effect point inferences, 01 = ((31,u1):
PLD[0l]{u) = 2[PL(-, y, 0) PL{lmgu{lltl)\ y,0lu,gp(Oiu))\, (5.29)
where 01)W = comes from = (/^,,/^J' and = {'lu,'2J and
is/p(0iu),^u{9iu)) maximimizes PL{lui,u2;y, ¡3lu, (32) with respect to (I32,u2). For
linear mixed models, the acceleration matrix follows from the arguments for deriving
the acceleration matrix of LD^(uj) in Section 4.6.4, and is given by
M = -2A '(h -
0 0
-l
0 h
22
)A,


CHAPTER 2
COMPUTATIONAL ISSUES IN LOCAL INFLUENCE ANALYSIS
In this chapter, computational formulas for local influence are reviewed
(Cook 1986; Beckman et al. 1987), To demonstrate the mechanics of the formulas,
detailed derivations of previous results for the regression model (Cook and Weisberg
1991; Schwarzmann 1991) will be presented with minor additions. Next, the issue
of how to obtain objective benchmarks is considered, and some new novel solutions
are outlined. Finally, local influence is discussed under reparamaterizations of the
perturbation space fi and the parameter space 0. By applying previous results
(Escobar and Meeker 1992; Pan et al. 1997), some new diagnostic tools for influence
on fitted values in linear regression are derived.
2.1 Curvatures and the Direction of Maximum Curvature
Cook (1986) showed that the curvature of LD(u¡) in direction v can be
expressed as
Cv = 2v'A'(-L~1)Av, (2.1)
where L is the observed information matrix of the original model,
d2m y)
L =
dOdO'
6=0
and Apx? has elements
d2K(0-y)
dOiduj
1
0=0,(jJ= for = 1,... ,p and j = 1,... q. This result is proved in Chapter 4 as a special case
of Theorem 4.2.
21


LOCAL ASSESSMENT OF PERTURBATIONS
By
GLEN LAWSON HARTLESS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2000


64
Cmax
40r
35
30
25
20
15
10

5

10
15
20
Row
Figure 3.4: School data; maximum curvatures for single row perturbations
perturbation presented in Section 3.4. Note that the curvature and directional
vector change if one or more of the columns of X are rescaled.
3.3.4 Influence on f3 and 9
The acceleration matrix for directed influence on ¡3 is
hu0p + rl(X'X)-1 rx¡3x[(X'X)~x n{X'X^x^'] (3.17)
The eigensystem of this matrix is unknown, but my numerical examples indicate
that it has k distinct non-zero eigenvalues. The eigensystem of F for the full set of
parameter estimates is also unknown.
3.3.5 Example
The values of school 18s regressors are most influential when perturbing rows
of X one at a time, as evidenced by a large maximum curvature (Figure 3.4). For
perturbing row 18, vmax is nearly proportional to ¡3 (Figure 3.5). Note that although
element 3 of vmax appears quite large, it is actually less than the benchmark of


143
are usually written
X'RlX X'RlZ
Z'RlX Z'RlZ + Gl
X'Rly
Z'R'y
(5.17)
The corresponding set of estimating functions are
(5.18)
Z'Rl(y -X0- Zu) G~lu
5.4.4 Inference by Profile Predictive Likelihood
The BLUEs and BLUPs can also be derived by the profile predictive
likelihood. Recall that inference using the profile predictive likelihood works as
follows:
1. Maximize the joint likelihood of u and /3 with respect to /3. Denote
the solution as /3(y, u).
2. Substitute ¡3(y, u) into the joint likelihood. The resulting expression,
L*(u, (3(y,u)-,y), is the profile predictive likelihood of u.
3. Obtain a prediction for u by either of the following methods:
a. Obtain the maximum likelihood predictor (MLP) by max-
imimizing L*(u, /3(y, u); y) with respect to u.
b. Obtain the conditional/predictive mean of u given (y', /3 (y, it))
from /i(u|y,j8(y,)) oc L*(u, ¡3{y, u); y).
4.Obtain point estimates of /3 by substituting the predictor of u from
step three into $(y,u).
By definition, the point inferences from steps 3a and 4 maximize the joint
likelihood of u and 0. This implies that using the MLP from the profile predictive
likelihood leads to the BLUEs and BLUPs. However, it is not as clear that the
solutions from step 3b and 4 lead to the same solutions, and so the derivation


have
69
= J¡ [(jtr-j8x) ((X'jr)-1r'-^'(Jr'X)-1jr')] (Vj r)
2
<7
- + WPf)^! r)
O Oj
= ^\(hr-px)
A.
y2^ + m2)^jr)
(3.26)
2 nirll2
tj [r- ¡3 X(fij)
|(^ + \0\?)(v¡ r)
-\\Pf(Vir)-'f-0Xv¡)
The following matrix multiplication yields the same expression as (3.26):
,m-2(^ + ||V) 0xVj)
= 3 [(/* r 0 x) ((X'X)"1 r' $' (X'X)-X')] 0 XVj)
2 llrll2 .
2
a2
2
= 55 [(J r p x) (0 ||0|| Vj)] T5(M- + ||/3||2)03 X,)
= r + 0xVj) ~ + ll/8||2)0 XV,)
-\W( j
Using (3.26) and (3.27), confirmation of the jth eigensolution is immediate:
(Fm \jI)Vj = 2(j^~ + !|£)/ ] (Vk_i+1 r)
^+ I 0 ***-*>
(3.27)
(3.28)
= 0.


BIOGRAPHICAL SKETCH
Glen Hartless was born in Summit, New Jersey in 1971. As he grew up, Glen
always had an interest in keeping the numbers. While researching careers in
applied mathematics in the tenth grade, he stumbled upon the statistics profession
and knew that it was perfect for him. (Considering the content of his dissertation, it
is ironic that frustration with the abstractness of matrices is what led him to the
field.) Interestingly, while he was still in high school, his father arranged for him to
meet Daryl Pregibon, a pioneer in developing diagnostics for logistic regression.
Perhaps this chance meeting influenced Glen more then he knows. As an
undergraduate at Virgina Tech, he completed a thesis on nonparametric density
estimation. Glen also studied exploratory graphical techniques for analyzing
reliability data while at an NSF-sponsored Research Experience for Undergraduates
(REU) program at the College of William & Mary. After receiving his B.S. in
statistics from Virginia Tech in 1994, he attended the University of Florida in
pursuit of his Master of Statistics and Ph.D. degrees. There, he did collaborative
research for four years in the Department of Family, Youth & Community Sciences.
Glen also met and married his wife, Christine, in Gainesville.
202


57


0.75
4
0.5
J
0.25
2
,
... i


5 10 *15 20
0.25



5 10 *15 20
.


-0.5
.

(a) Residuals
(b) Externally stu- (c) Cooks Dis-
dentized residuals tances

2
2
o.:
1.5
l.S
0.4
.
1
1
0.3



0.5
.... .
V
.
0.5
. .
0.2
i |

0.5
*s
ft
'ft ..ft
0.5
5 ft ft 20
0.1


1
-1
.
.
1.5
1.5
5
10 15 20
-2
2
(d) Hat diagonals
(e) DFBETAS for
SALARY
(f) DFBETAS for
WCOLR
1.5
1
0.5
1
1.5
2
(g) DFBETAS for
SES
2
5
1
5
5
1
5
2
S 11 t 8 20
(h) DFBETAS
for TSCORE
2
1.5
1
0.5
1
1.5
-2
(i) DFBETAS for
MOMED
Figure 3.1: School data; diagnostic plots


56
Table 3.1: School data; Pearson correlation coefficients
SALARY
WCOLR
SES
TSCORE
MOMED
VSCORE
SALARY
1.00
WCOLR
0.18
1.00
SES
0.22
0.82
1.00
TSCORE
0.50
0.05
0.18
1.00
MOMED
0.19
0.92
0.81
0.12
1.00
VSCORE
0.19
0.75
0.92
0.33
0.73
1.00
level (MOMED). Appendix A contains model fitting information and diagnostics for
the regession analysis, with all variables centered and scaled.
Diagnostic plots of the residuals, externally studentized residuals, Cooks
Distances, diagonal elements of the hat matrix, and DFBETAS appear in Figure
3.1. Ordinary least squares regression indicates cases 3 and 18 are outliers, although
Huynh (1982) also identified observation 11 as a mild outlier. This data set also
has some redundancy among the predictors SES, WCOLR and MOMED (Table
3.1). Although the X matrix is not ill-conditioned, the association between SES
and MOMED accounts for the unexpected negative slope estimate for MOMED.
Table 3.2 shows the non-zero curvatures (eigenvalues) for perturbing
single columns of the X matrix. Comparing the columns of the table provides
information on which regressors measurement is most influential. Here we see
that the measurement of SES is typically the most influential, both for regression
parameter estimates and for a2. Comparing the rows provides information on which
parameter estimates are most influenced by these kinds of perturbations. Here we
see that WCOLR and MOMED estimates are the most sensitive. These are the two
regressors that are highly correlated with each other and with SES. It comes as no
surprise that the results for collinear explanatory variables are more susceptable to
measurement errors of this nature.


199
Poon, W.-Y. and Poon, Y. S. (1999), Conformal Normal Curvature and
Assessment of Local Influence, Journal of the Royal Statistical Society,
Series B, 61, 51-61.
Prentice, R. L. (1988), Correlated Binary Regression With Covariates Specific to
Each Binary Observation, Biometrics, 44, 1033-1048.
Rancel, M. M. S. and Sierra, M. A. G. (1999), Measures and Procedures for the
Identification of Locally Influential Observations in Linear Regression,
Communications in Statistics, Part A Theory and Methods, 28, 343-366.
Robinson, D. L. (1987), Estimation and Use of Variance Components, The
Statistician, 36, 3-14.
Schall, R. and Dunne, T. T. (1992), A Note on the Relationship Between
Parameter Collinearity and Local Influence, Biometrika, 79, 399-404.
Schall, R. and Gonin, R. (1991), Diagnostics for Nonlinear Lp-norm Estimation,
Computational Statistics and Data Analysis, 11, 189-198.
Schwarzmann, B. (1991), A Connection Between Local-influence Analysis and
Residual Diagnostics, Technometrics, 33, 103-104.
Searle, S. R. (1971), Linear Models, New York: Wiley.
Searle, S. R. (1982), Matrix Algebra Useful for Statistics, New York: Wiley.
Searle, S. R., Casella, G., and McCulloch, C. E. (1992), Variance Components, New
York: Wiley.
Shi, L. (1997), Local Influence in Principal Components Analysis, Biometrika,
84, 175-186.
St. Laurent, R. T. and Cook, R. D. (1993), Leverage, Local Influence and
Curvature in Nonlinear Regression, Biometrika, 80, 99-106.
Swokowski, E. W. (1988), Calculus with Analytical Geometry (2nd ed.), Boston:
PWS-Kent.
Tang, M. K. and Fung, W. K. (1996), First Order Local Influence of Test
Statistics in Multivariate Regression, Sankhya, Series B, Indian Journal of
Statistics, 58, 323-337.
Tawn, J. (1987), Comments on Parameter Orthogonality and Approximate
Conditional Inference, Journal of the Royal Statistical Society, Series B, 49,
33-34.
Thomas, W. (1990), Influence on Confidence Regions for Regression Coefficients in
Generalized Linear Models, Journal of the American Statistical Association,
85, 393-397.


182
in county 12, the countys predicted mean hectares of corn per segment is
affected (Figure 5.4(b)). Using the weighted county perturbation scheme leads
to inconclusive evidence about county 12s overall influence. Although element
12 figures prominantly in the umaxs for the two slope estimates (Figure 5.6),
even the 12th countys predicted mean appear unaffected (Figure 5.8). Influence
diagnostics for perturbing cluster variances (Figures 5.9, 5.10, and 5.11) suggest
that the assumption of equal variance for the 12th random effect is not influential.
Therefore, it is deemed that differential weighting of observation 33 is the main
cause of sensitivity, and since the value was considered suspect a-priori, deletion or
substantial down-weighting of this single observation seems warranted.


CHAPTER 4
LOCAL ASSESSMENT FOR OPTIMAL ESTIMATING FUNCTIONS
It is possible to examine the local influence of perturbations on measures other
than the likelihood displacement, LD(u)), and the profile likelihood displacement,
LD^6l\u>). Several authors have applied local influence techniques to specific
measures of influence. For instance, Lawrance (1988) and Wu and Luo (1993c)
looked at influence on the Box-Cox transformation parameter. Also, influence
has been examined on the residual sums of squares in regression (Wu and Luo
1993b), deviance residuals and confidence regions in GLMs (Thomas and Cook 1990;
Thomas 1990), and goodness-of-link test statistics in GLMs (Lee and Zhao 1997).
However, considerably less attention has been given to assessing local influence
when estimation is not performed by ML. Some work has been done on extentions
to Bayesian analysis (McCulloch 1989; Lavine 1992; Pan et al. 1996), and Lp norm
estimation (Schall and Gonin 1991).
In this chapter, local influence techniques are extended in two ways. First,
we consider the influence of perturbations on a general class of influence measures.
Second, we address parameter estimates that are found via estimating equations.
The intent is to unify nearly all previous work on this methodology and provide a
conceptual basis and computational tools for future applications. The first three
sections outline the inferential platform, a general class of influence measures,
and assumptions about the perturbations. Next, techniques for local assessment
of perturbations are discussed. In Section 4.4, we give some algebraic results
for the two main diagnostic quantities: the gradient vector and the acceleration
matrix. Section 4.5 contains computational formulas, while Sections 4.6-4.8 contain
applications. Several measures, including LD( 87


76
B1
(a)Influence curves for 0x .2,
02 = A
B1
(b)Influence curves for 0x .2,
02 = -7
B1
B1
(c)Influence curves for 0\ = .5, (d) Influence curves for 0x = -5,
02 = -4 02 .7
Figure 3.6: Effects of single and double column perturbations on ¡3\. Solid line-
double column; dashed line- column 1 only.


29
Finally, the acceleration matrix is
F = 2
X hr
a2 {xx'y1
o'
0
2ct4
n
O2
X'
V
G
(2.10)
2 4
= ^ff + -4rrr'.
ncn
2.4.2 Directed Influence on a1
The acceleration matrix for the profile likelihood displacement for a1 is
simply the second term of (2.10):
(2.11)
r 2l
Solving the characteristic equations (.r AI)v = 0 directly, the single
eigensolution is
T'max Amax ^max OC T.
Therefore, as mentioned in Chapter 1, vmax for directed influence on o2 is
proportional to the residuals,
2.4.3 Directed Influence on /3,
Without loss of generality, consider directed influence on Rather than
find the acceleration matrix, we proceed directly to determining the sole eigenvector
proportional to A'T and the corresponding curvature 2||A'T||2, as per (2.3) and


129
The direction of maximum local influence is then
Umax a D(y fi)Xh 1X'z1.
Plotting the elements of t?max against case indices indicates that elements 24 (site
3 variety 4) and 65 (site 7 variety 5) are influential (Figure 4.4). When plotting
elements against the corresponding linear predictor, it is clear that observations
with moderately-sized proportions are more influential (Figure 4.5).
As in Wedderburn (1974) and McCullagh and Nelder (1989), we consider
fitting a model with an alternate variance function, V(/u) = ^2(1 A*)2- This function
reflects the assumption that extreme proportions are more precise than those arising
from a binomial distribution. Pearson residuals indicate that the alternate variance
function is more realistic, but that a relationship between the residual variance
and the the linear predictor remains (Figure 4.6). Implementing the case-weight
perturbation and assessing its effects on the Pearson Chi-square statistic requires
the following quantities:
A
y- A
A(i A)
),
and
dm
= X'z2,
where the n x 1 vector z2 has ith element
2(A? ~ 3i¡yi + + yt) y?)
Ai(l Ai)2
The direction of maximum local influence is then
oc D(


163
5.8.3 One-way Random Effects Model
Using (5.30), umax for directed influence on a linear combination of the
intercept and random effect point inferences is given by
Umax OC A'/l X* OC
oc
D{ei)'R-lX 0gxg
D(ei)'ln 0gxg
£"=i eu
o
ffXff
OC
E%i ^
£j=i eU
E£i egj
h
X
hn
hU
h21
h22
A11
h12
hn
h22
X
X
hn h
12
X
e9j
Using the relationship between c and i given in (5.32),
Cij TliCi 71 ll 71 j
7=1
Vj? + cr2/n
Ui I TliUi ,> Hi
(5.47)
(5.48)
This implies that the direction of maximum curvature is proportional to , the
vector of BLUPs. Thus, we see that the form of A has an interesting consequence
when the intercept is the only fixed effect in the model: no matter what linear
combination is chosen, the direction of maximum curvature is the same. This
anomaly does not occur when /3 has more than element.


2
Following Cook (1986), a vector of q perturbations is denoted as
oj = ,Wq)', and the ^-dimensional space of possible perturbations is
denoted as fi. Included in this space is one point, uio, that does not change
the original formulation of the problem, and this is called the null perturbation.
The perturbations are not parameters to be estimated, but rather they are
known real-valued quantities introduced into the problem by the analyst. These
perturbations might represent measurement errors, unequal variances, case weights,
correlated errors, missing predictors or some other change to the data, model or
inference method. Prudent choice of perturbations can lead to useful diagnostics.
For example, if parameter estimates are sensitive to perturbing the contribution
(weight) of a particular observation to estimation, it indicates that the observation
is overly influential on the estimates.
For data y and parameters 0, let L(0; y) denote the log-likelihood from
the original postulated model, and let Lu(0\y) denote the log-likelihood resulting
from the inclusion of perturbations. If the perturbation specified is the null
perturbation, then the two likelihoods are equivalent. The resulting maximum
likelihood estimators (MLEs) of these two likelihoods are denoted as 0 and 0u,
respectively. Cook (1986) used the likelihood displacement to summarize parameter
estimate changes under perturbation:
>M = 2[(;w)-£(d;y)|. (1.1)
From inspection of (1.1), we see that the likelihood displacement is the difference of
the original likelihood evaluated at two values: the original MLEs and the perturbed
MLEs. Because 0 maximizes the original likelihood, LD(w) is non-negative and
achieves its minimum value of zero when u> = w0. As the perturbed MLEs become
increasingly different from the original MLEs, LD(w) becomes larger. Thus, the


27
Using these results, we find
1
T =
a
(2.9)
~7i
0
and this leads to the following expressions for the maximum curvature and its
direction when directing influence on
umax oc AT oc A'Pl A^(i)7!.
The next two sections apply these results to document more completely the
two perturbation schemes given in Chapter 1.
2.4 Perturbations to the Response in Regression
The results of this section closely follow those of Schwarzmann (1991),
with only minor additions. As in Chapter 1, a set of additive perturbations are
introduced into the observed data to examine the influence of measurement error in
the response. Annxl vector of perturbations is introduced such that the perturbed
likelihood is based on Y + w ~ N(X(3,a2In). The null perturbation is u>0 = 0.
Equivalent results can be obtained by perturbing the model of the mean to be
X/3 cj. That is, by considering a model assuming Y ~ N(X(3 u>, a2In). Hence,
the measurement error perturbation can be interpreted as a type of mean-shift
outlier model; each perturbation is a stand-in for an additional parameter that
could correct or exacerbate lack-of-fit.


177
(a) vmax
i.-
(b) |t)max| star-plot
(c) v2
(d) |V21 star-plot
(e) v3 (f) |t>3| star-plot
Figure 5.5: Corn data; Scheme 2; local influence on 6


41
estimates of the redefined parameters can then be examined. Using similar steps to
Pan et al. (1997) and Escobar and Meeker (1992), we can write update formulas
for A^ and 1 and then calculate directed influence on a subset of the redefined
parameter estimates. Using partial differentiation, the building blocks of the
acceleration matrix under the parameterization 4> can be written:
A 90\
= d$A
r1 = * (d(f>Le d(f>
_ 90 80' x
6
(2.21)
(2.22)
where all partial derivatives are evaluated at the fitted model. From this, the
acceleration matrix for directed influence on (j)x can be obtained from the partition
4* = (^lJ (t>2)':
where
^11 = -2A'L? B+JA*
= _2A' (L~l-B ) A
2Aed^L* B^ d(f)Ae
= -2A )A
(2.23)
B22
0 0
0 L~
Lfa comes from the partition
L(j>
Lu
L 12
^'^>22
and all partial derivatives are again evaluated at the fitted model.
Although Pan et al. (1997) suggested that (2.23) is invariant to the choice
of 4>2, they did not provide a proof. To show this result, consider another


146
Meanwhile, substituting the solution from the first set of MMEs into the
second set yields
Z'R-lX{X,BTlX)-1X'R-x(v Zu) + {Z'R7XZ + G~1)u = Z'R ly. (5.23)
Solving for u provides the BLUP of the random effects,
= (Z'PRXZ + Gx)-lZ'P'Rly,
and this is identical to the predictive mean (5.22). Hence, predicting u with the
predictive mean from the profile predictive likelihood yields point inferences for u
and ¡3 that are identical to those from the mixed model equations.
5.5 Local Influence Methods for Linear Mixed Models
5.5.1 Assessment in the Marginal Model
Local influence techniques for linear mixed models have been developed using
the marginal model (5.13). Beckman et al. (1987) considered a special case of the
linear mixed model known as the variance components model. This model is used
for data with a sources of grouping or clustering, where the ith source has levels.
For Zi being of size n x and being of size gt x 1, the model is given by
y = X(3 + Zii + Z2U2 + ... Zaua + e. (5.24)
This structure implies Z is a block-diagonal matrix with ith block Z, and u has a
total of P2 = 5^=1 gi elements. In addition to the general LMM assumptions, the
vectors are mutually independent, with V(m,) = o\I9i and V(e) = o2In. The model
implies that the marginal variance of Y can be written as
a
V = ZGZ' + R=Y_] ofZiZ'i + a2I.
i=1


33
Second partial derivatives are then
d(5o *L<*> = 1 y'D(),
d and evaluating at the original model yields:
&L{0-y)
d/3du'
d2K{0-y)
1
= t1X'D( r)
0=6,tjj=u?o
da2duj'
1 w_ 1 ,
2d-4 r r 2d47*49
0=0, where is the denotes the Hadamard (element-wise) product, implying that rsq is
an n x 1 vector with elements rf. Combining these we can write
A' =
d2
D(r)X £r
and
F = 2d2
4f(r)X
+
4d4
n
(XX')"1
4^8 rst¡rsq
-^X'D(r)
(2.15)
a 9
o
D(r)HD(r) +
2nd2 rsqTsq
2.5.2 Directed Influence on d2
The acceleration matrix for the profile likelihood displacement for d2 is
simply the second term of (2.15):
>2] 1
* = r^rsnr
no
4 * (2.16)
Solving the characteristic equations directly, the single eigensolution is
r nr-n2
'-'max n.4
^max sq


96
eigenvector is the direction of maximum approximate effect. Other eigensolutions
can also be used.
4.3 Acceleration, Curvature and Normal Curvature
Two more geometric descriptions of the influence measure can be calculated:
curvatures and normal curvatures. These quantities are functions of both the
gradient vector and the acceleration matrix. When the gradient vanishes, the
curvature, normal curvature, and the absolute value of the acceleration associated
with a directional vector are all equivalent. From (1.5) the curvature associated with
direction v at the null perturbation is
i d2m{a) i
q 1 da2 I
(1 + (t?)2)3/2
Using (4.16), it follows that
^ \v'Mv\
C= (I+
(v'(Iq + rhrh')v)3/2
If the slope is zero, then C = |A|. As mentioned previously, curvature is a measure
of how quickly a curve turns.
The notion behind normal curvature is a bit more complicated. The vector
(wv(a)', 0)' in 5?9+1 is projected onto the tangent plane of the influence graph rather
than the graph itself. A plane is then created that contains both the projection
as well as the vector that is orthogonal to the plane, the principal normal vector.
This plane then intersects the surface to create a normal section. Using the tangent
plane as the basis, curvature of the normal section can be computed as if it were
a simple plane curve. The resulting normal curvature is a description of how the
influence graph deviates from its tangent plane. The normal curvature associated
'rhrh'v)3/2
\v'Mv\
(4.19)
(4.18)


APPENDIX A
MODEL FITTING INFORMATION FOR SCHOOL DATA
NOTE: No intercept in model. R-square is redefined.
Dependent Variable: VSCORE
Analysis of Variance
Sum of
Mean
Source
DF
Squares
Square
F Value
Prob>F
Model
5
17.21982
3.44396
29.019
0.0001
Error
IS
1.78018
0.11868
0 Total
20
19.00000
Root
MSE
0.34450 R-square
0.9063
Dep Mean
-0.00000 Adj R-sq
0.8751
C.V.
-6.745558E16
Parameter Estimates
Parameter Standard T
for HO:
Variable
DF
Estimate Error Parameter=0
Prob > |T|
SALARY
1
-0.139993 0.09301762
-1.505
0.1531
VC0LR
1
0.194122 0.22907761
0.847
0.4101
SES
1
0.919607 0.14859782
6.189
0.0001
TSC0RE
1
0.250736 0.09464627
2.649
0.0182
M0MED
1
-0.203697 0.22031344
-0.925
0.3698
Variance
Variable
DF
Inflation
SALARY
1
1.38519879
WC0LR
1
8.40130895
SES
1
3.53513942
TSC0RE
1
1.43413046
M0MED
1
7.77076288
Collinearity Diagnostics
Condition Var Prop Var Prop
Var Prop
Var Prop Var Prop
Number Eigenvalue
Index SALARY WC0LR
SES
TSC0RE M0MED
1 2.83681
1.00000 0.0134 0.0129
0.0296
0.0071 0.0142
2 1.39508
1.42598 0.2192 0.0040
0.0020
0.2456 0.0026
3 0.49664
2.38997 0.7601 0.0006
0.0064
0.6485 0.0004
4 0.20253
3.74262 0.0011 0.0655
0.9471
0.0418 0.1258
5 0.06894
6.41490 0.0062 0.9170
0.0149
0.0570 0.8570
188


93
eigenvalues are proportional to the lengths of the principal axes of a confidence
ellipsoid for 0. Hence, the determinant of the variance provides a single measure
of the confidence ellipsoids size. Provided the estimating functions have been
standardized, m% is the log of the estimated generalized asymptotic variance of 0. It
is a function of u> both directly and indirectly via 7(0^) 0u.
As a final note, it is not necessary for the influence measure to be a function
of point estimators. In Bayesian analyses, perturbation can be measured by
the Kullback-Leibler distance between the perturbed and unperturbed posterior
densities. Such a measure would depend on w, but not on any point estimates of
0. McCulloch (1989) developed a general local influence technique for assessing
perturbation of the prior distribution in a Bayesian analysis. Other types of
perturbation analysis for Bayesian models are found in Kass et al. (1989), Geisser
(1991), Geisser (1993), and Weiss (1996).
4.2 Local Assessment of Perturbations
At this point we have set the stage for our sensitivity analysis: a perturbation
is introduced into the inference, and a measure of the effects is chosen. The next
step is to obtain some useful diagnostics. This section decribes how a Taylor series
expansion of m and some differential geometry results can be used to describe the
function.
4.2.1 First-order Local Assessment
As in Chapter 1, let u)v(a) = u>0 4- av be a vector in the space of
perturbations. The unit-length vector v determines the direction from u?o and the
value a determines the distance from cjq in fi. The value of m for a perturbation of
length a in direction v can be approximated by a first-order Taylor-series expansion


97
with direction v at the null perturbation is (Cook 1986)
NC =
\v'Mv\
(1 + \\rh\\2yi2v' (I q + rhrh')v
(4.20)
If the gradient is zero, then NC C = \A\.
The two geometric descriptions NC and C can be distinquished as follows.
Curvature is the reciprocal of the radius of the best fitting circle to the lifted line
associated with v, while normal curvature is the reciprocal of the radius of best
fitting circle to the normal section associated with v. If the gradient is zero, lifted
lines are normal sections, implying that NC = C = |A|. Additional insights are
found in Cook (1986), Wu and Luo (1993a), Wu and Luo (1993b), and Fung and
Kwan (1997).
4.3.1 How to Use Local Assessment of Perturbations
Local assessment of perturbations can be used to (1) assess the influence
of aspects of the statistical problem on inferences, and (2) compare the relative
sensitivity of parameter estimates.
First, we define the local influence associated with a length-one vector v to
be:
ilm'vl if to 0
(4.21)
\v'Mv\ if rh = 0.
Thus, li(v) is simply the directional derivative unless the gradient vanishes, in which
case it is the normal curvature. The direction of maximum local influence, umax, is
the main diagnostic tool, and it can be displayed graphically using an index plot or
star plot to informally assess the influence of perturbations. Additionally, the local
influences of basic perturbations can be plotted against case indices to compare the
effects of one-at-a-time perturbations.


73
are orthogonal. The implications of these simplifications are as follows:
1 -PI-ft
llvll= IMI = ll^all = i
h = *i y
Si = 1 = 1
The first two eigenvalues of F are the same:
r'r
A
x2y
h-
r , \m\ 1+ # + &
Cmax Ai = A2 2(n + ~^r) 2nx
and we may choose either of the following vectors for i>max:
r-hXi
-PiX2
-PiXx
r p2X2
Without loss of generality we choose the first one. Letting Xu denote the perturbed
design matrix, we have
Xu = X+aW
= X +CL
(3.32)
r fcXx -P2X1
ar + (1 a¡3i)Xi X2 X1 -^2,u>
for some real number a. Now, consider the effects of perturbation on fi\. Assuming
a no-intercept model is still fit after perturbation, the perturbed estimate is given by
r[,uy
P\,u
Fl,a
2
(3.33)


119
4.7.2 Variance Perturbation for ML Regression
In this section, first-order local assessment diagnostics are derived for the
variance perturbation in ML regression with the perturbed generalized variance
as the objective measure. Because h has a simple form, we derive su and s1
element-by-element rather than construct the matrices in (4.54) and (4.55).
4.7.2.1 Construction of s^
To derive the ith element of su we begin with the following results from
Section 2.5:
d2Lu(0-,y)
dflduji
d2K(e,y)
da2du>i
= -^(yi xfflxi
(Vi x'i0)2
2 (4.61)
(4.61a)
where Xi is the covariate vector for the ith observation. Taking partial derivatives of
(4.61) leads to
&Lu(0-,y)
dpdp'dwi
d3Lu(0-,y)
dpdo2dui
Avi x'iP)xi.
xi(pl)xi
Taking partial derivatives of (4.61a) yields
a3(9 ;y)
d{o2)2dui
--(y.-xrpf.
After evaluating these expressions at the null perturbation, we can construct the
matrix
dhUJ(0: y)
&Lu(0;v)
2X{Xi Q*Xi
divide'
0 dOdO'duii
0
-£x'i ~i
(4.62)


Ill
Construction of the gradient and directional derivative is immediate:
S = rh'v
= , + siKJ)v
= x'L 1 At;.
Note that the direction of maximum absolute slope is proportional to vh oc A'L lx.
This is exactly the same as the direction of largest curvature for the profile likelihood
displacement of x'Ow under the orthogonal parameterization given in Theorem 2.1.
4.6.2 Ratio of Estimates
For the measure m = 9^/9iw, the building blocks for a first-order local
assessment are
7(0u,)= L K= <
S(J = 0 S-y
where is a p x 1 vector with a 1 in the ith position and zeroes elsewhere. Hence,
S = rh'v = (si + siKJ)v = Av = V'Av,
O 9j
where ' is the ith column of L \
4.6.3 Likelihood Displacement
Recall that the likelihood displacement is given by
m(w, 7(0o,)) = 2[L(0; y) L(0U-, y)].
This influence measure requires a second-order local assessment, as is now shown.
First, it does not depend on uj explictly, so sw vanishes. Since the exact form of the


CHAPTER 6
MULTI-STAGE ESTIMATIONS AND FUTURE RESEARCH
In this chapter, several areas of future work are outlined, most of which stem
from the generalization of local assessment to estimating functions in Chapter 4.
First, some preliminary results are given for extending the technique to multi-stage
estimations. Then, a synopsis of the dissertation is presented along with some
additional areas of future research.
6.1 Local Assessment in Multi-stage Estimations
Sometimes inference takes place in more than one formal stage. One example
is linear mixed models when variance components are unknown. Restricted/residual
maximum likelihood (REML) (Patterson and Thompson 1971; Robinson 1987;
Searle et al. 1992) can be used to estimate the variance parameters. Then the mixed
model equations are used to estimate ¡3 and predict u, with the estimated variance
parameters treated as known. Local influence techniques can be used to assess the
effects of perturbations on functions of the two sets of parameter estimates using
results from Chapter 4.
First, let 0X denote the solution of p\ estimating equations hx(9i;y) 0.
Further, let 02 be the solution of p2 estimating equations h2(92,y,9i) = 0 for
fixed 9\. Perturbation leads to two sets of perturbed equations, /ii,w(0i;y) = 0
and h2^(92;y,9itU) = 0, with solutions 9\and 02,u respectively. The effects
of perturbation are measured by ra, which depends on u; explicitly and via
and 92)W. The algebraic expressions and computational formulas for first- and
second-order influence given in Chapter 4 can be used by simply using the notation
9' = (9[,9'2).
185


138
density, the predictive mean, is the preferred point predictor. Standard errors and
prediction intervals can also be constructed from this predictive density.
5.2.2.1 Example
Consider again the situation of predicting future observations in a linear
regression model with O' = (f3',cr2) unknown. Naively replacing 0 by 6 in the
conditional distribution of u given y yields the estimative predictive likelihood.
Hence, inference proceeds under the assumption that u is conditionally distributed
as normal with mean Z¡3 and variance d2Im. A prediction interval for a single u
with covariate vector z would then be
z'¡3 Za/2 In contrast, the conditional distribution of u from standardizing the profile
predictive likelihood is multivariate t with mean Z/3 and dispersion parameter
na2(I + Z'(X'X)~1Z), as shown by Levy and Perng (1984). A prediction interval
for a single u would then be
z'0 tn_payj 1 + z'{XX)-iz.
This matches the exact prediction interval given in (5.5).
5.3 Influence Measures for Prediction
In this section, some general measures are proposed for assessing the effects of
perturbation on point estimation and point prediction. Notation is as follows: $ are
point estimates for parameters (3 and are point predictions for random variables
u. Also, let ¡3W and u denote the perturbed versions of these point inferences, and
let PL(u; y,(3) denote the log of the unstandardized predictive likelihood.


135
larger than that of :
MSEP() = V(u)
= E(V(u|)) + V(E(|y))
> E(V(u|#))
= E(E((u -)2|s/))
= E(CMSEP())
= MSEP(w).
(5.1)
Other predictors can be derived if the class of predictors is restricted. For
instance, minimizing MSEP among predictors that are linear functions of y results
in the best linear predictor (Searle et al. 1992):
BLP(u) = E() + CV-'iy E(y)),
(5.2)
where V is the variance of y and C is the covariance of u and y. Although
not required by its definition, the BLP is an unbiased predictor of u since
E(BLP{u)) = E().
5.1.2 Prediction After Data Are Observed, but with Unknown Parameters
Typically the goal of prediction is accompanied by a need to estimate
unknown parameters 6. Here, we may first estimate the parameters using the
observed data and then estimate the best predictor. The MSEP of the estimated
best predictor can then serve as a measure of uncertainty in the prediction, although
Booth and Hobert (1998) argue that CMSEP is more relevant.
To illustrate, consider the normal linear regression model with unknown
parameters, /3, but with known variance, a2. After observing data y, the goal
is to predict future independent observations u ~ N(Z/3, a2Im). From either
ordinary least squares or ML, = (X'X)~1X'y. Hence, the estimate of


154
Of particular interest are the first and second columns of h
-i
h~d' t
a2+a2/ni
h ld2
a2a2c
k(a2 + cr2/ni)
-1
or2 i k_
a2
a2+ (T2
t2 4- (5.36)
(5.37)
5.7 Residual Variance Perturbation Scheme
In this section, a residual variance perturbation is proposed. The perturbation
is applicable to the general form of the linear mixed model. Simplifications when
R = a21 are presented, and some closed-form expressions for the one-way model are
given.
5.7.1 Form of the Perturbation
Let Cij denote the ijth element of the matrix R. A set of n perturbations is
introduced into the model such that the perturbed residual covariance matrix is
£11
Wl
<712
<713
y/U 1U>3
<7ln
y/UlUn
<712
£22.
<72n
y/UJ 1U/2
<713
U>2
y/UJ2(jJn
<73 n
y/U 1U>3
.
y/U3U)n
CTln
<72 n
<7 71
y/Unur,
y/U2Un
V'w3Wn
<*
The intent of this scheme is to weight the contribution of individual observations
to inference in situations when they do not necessarilly enter the likelihood in an


59
i
0.75
0.5
0.25
0.25
-0.5
-0.75
-1

,
r!- A *~ft 2o
t
Figure 3.2: School data; X3 + w perturbation; influence on $
3.3 Perturbations to a Single Row
The special case of only perturbing a single row of X was discussed in
Thomas and Cook (1989) in generalized linear models. Here, detailed derivations
for regression are presented. The results indicate that single row perturbations have
limited diagnostic value.
3.3.1 Building Blocks
Without loss of generality, suppose perturbations are added to the k regressor
values of the first observation (i.e., x\ becomes x\ + u>'). For d\ being of order n,
the perturbed likelihood is
M; y) .. k k (3.13)
= -|iog j=i j=i
Partial derivatives with respect to a? are as follows:
(3.14)


62
3.3.2 Directed Influence on a1
The acceleration matrix for the profile likelihood displacement for a1 is given
by
*"-*[-#] I v
4 r\
41 r
(3.16)
ncr*
100
Solving the characteristic equations (F[a XI)v directly, the single eigensolution is
Cmax ^ll^ll ^max 0-
Notice how the curvature depends upon the residual for the case that is being
perturbed, but the directional vector is always proportional to 0 regardless of which
row of X is being perturbed. This suggests that rmax is not very useful for single
row perturbations. The values of Cmax for each row can be compared to assess which
row is most influential when perturbed. However, the relative effects only depend
upon the residuals, indicating there is little value added by plotting the curvatures.


98
The relative sensitivity of p parameter estimates can be assessed by plotting
the p maximum local influences for mi = 9i^,..., mp = 6pu. Measures with the
largest maximum local influences are most sensitive to the perturbation. Two
cautions with this technique should be noted. First, the metric of the parameter
estimates ought to be comparable. If the metrics are not comparable, standardized
parameter estimates should be used. The second caution pertains to first- and
second-order influence. Typically, the set of p measures are assessed by a first-order
approximation. However, if a subset of the measures requires second-order
assessment, then the following revised definition can be used to make the local
influences comparable:
li*(v) =
| rn'v\ ifm^O
\~v'Mv\ if m = 0.
(4.22)
This ensures that all influence measures are compared by the approximate effects of
length-one perturbations.
4.4 Algebraic Expressions for Local Assessment
In this section, algebraic expressions for local influence diagnostics are
derived.
4.4.1 First-order Assessment
Theorem 4.1
The directional derivative in direction v of inuence measure rn(u), 7(6^)) at the null
perturbation is given by:
S=(s' + s'yKJ)v,
(4.23)


120
Accordingly, the trace of the following matrix multiplication is the ith element of
dK(G\y)
dundo'
II
i-H
1
-C
^ T 1- 7 .
0
^(X'X)-1
0
0
1
A
1
q4-i
1
1
O
2 d4/n
Xix'iXX)
-1 2r.
Xi
(4.63)
In order to find a formula for the trace, an expression for the kth diagonal element
of the upper left portion of the matrix is needed. Letting ajk be the jkth element of
(X'X)-1, the kth diagonal element of Xx\[X'X)*1 is
p-1
^ ] ^jkxijxik-
3=1
Hence, the trace of (4.63) is given by
p-1 p-1
-EE xijQjkxik) ~.2 0 x'j Yl***)) 2
k-1 j=1 j=1 k=l
From this, the vector s can finally be constructed:
(EL ES=l(i*ifc)) + 5
(5£i%£Gto***)) +
p-i,
2r2
lli
X21C2
^nl*
an
^12*i
2i*2
a21
i(p-i)
_ xn2X'n
_ a2(p-l) _
^(p-l^l
x2(p-l)x2
xn(p-l)Xn
a(P~ 1)1
a(p-l)(?-l)
' sq
no2.
(4.64)


44
Theorem 2.1
The maximum curvature and its direction for directed inuence on x'O is given by
r 2
Umax~ ifai
where ki = Le l^x.
Proof of Theorem 2.1
lIlAji,' x||2
Vr
oc A'eLe 1 x
I T 1/21
To begin, consider a model with parameterization

a: =
ki k2 ...
is a full rank pxp matrix with
, r-i/2
-Lj q X
k'ikj =0 i^j
k[ki= k[ki i = 2...p.
By construction,
1 = x'6
0 = tJ^L?/2K(I>
K =
do
IIM2
K'
1 l:1/2k.
,|fei||2 0 V d' ||fci||2 0
Using the update formulas in (2.21) and (2.22) it follows that
A _96|,A
" dj *
1 K'L~1/2Ae
IIM2
-1 / 96 Wlr-l (de> '.-i
L* ~ L* {dj}
= K'L1J2L¡1L10/2K = \\k1fl.


5
Figure 1.2: Curve with two circles of curvature
The reciprocal of curvature is called the radius of curvature, and it is the
radius of the best fitting circle at that point on the curve. These circles of curvature
have the same first and second derivatives and tangent line as the curve and lie on
its concave side. Figure 1.2 shows the curve (|sin(i), |sin(2f)) and two of its circles
of curvature. The one on the left occurs when t y- while the other one occurs
when t = arccos(O). Their curvatures are 6 and 8/9 respectively. The expression
turning on a dime refers to to a sharp turn: one in which the radius of curvature
is small, and consequently the curvature of the path is large.
The formula for curvature is
C =
f'(t)g"(t)-9'(t)f"(t)
((/W + foW)*
We are particularly interested in assessing curvature at the local minimum of a
function from x to y, i.e. when x = t. For this special case, curvature measures
how quickly the function increases from that minimum value. In this case, (1.5)
simplifies to the acceleration of the curve:
d2g{x)
(1.5)
C =
dx2
(16)
X=Xq


107
The theorem below now provides a formula for P that does not require
re-estimation of the model.
Theorem 4.4
Let 0^ be the solutions to the p perturbed estimating equations hu(6; y) = 0. Then,
P =
* -1 r
I
-h [ Iq J' ]
Tf
-
J
(4.41)
where
O' =
(4.42)
Proof of Theorem 4.4 Taking the p sets of equalities of the form given in
Lemma 4.2, we find
-hl Iqj P
-h'p /,] P
Iq J'
Iq J'
(4.43)
Manipulation of the left side of the equality proceeds as:
-h [ I q
hnIq hi2lq ...

P =
h>2\I q h22lq
P
-h'p Iq
: :
-h Iq
P


This dissertation was submitted to the Graduate Faculty of the Department of
Statistics in the College of Liberal Arts and Sciences and to the Graduate School
and was accepted as partial fulfillment of the requirements for the degree of Doctor
of Philosophy.
August 2000
Dean, Graduate School


58
Table 3.2: School data; curvatures using X* + u; perturbation
Estimates
Column being perturbed
in LD(w)
or LD^(u>)
SALARY
WCOLR
SES
TSCORE
MOMED
9
H-*
II
4*.
i1
4^
4x
Ax=20.156
Ax-50.008
Ax6.658
Ai=19.065
A2=0.440
A2=0.847
A2=19.002
A2=1.413
A200.932
A3=0.094
A3=0.071
A314.441
A3=0.599
A300.091
b
Ax=3.357
Ax=18.534
Ai=26.445
Ax-4.432
Ax17.292
A2=0.440
A2=0.847
A2=19.002
A2=1.413
A200.932
A
3.357
0.956
19.008
2.126
0.951
ft
0.458
18.534
19.890
1.585
10.869
ft
0.443
2.957
26.445
1.501
1.640
ft
1.129
1.859
19.220
4.432
1.295
ft
0.444
11.590
19.324
1.480
17.292
<72
0.881
1.693
38.004
2.825
1.865
Since perturbing the SES variable is most influential, examining directional
vectors for this perturbation is of interest. The element corresponding to case 18 is
the most noticeable in the plot of t>max for the likelihood displacement (Figure 3.2).
This observation has a large residual and is indicated as being influential in the
various diagnostic plots in Figure 3.1. Also note the similarity of Figure 3.2 to the
residuals (Figure 3.1(a)).
Plots (a)-(e) of Figure 3.3 show the directions of maximum curvature when
directing influence on each of the individual regression parameter estimates. In
general the plots indicate different elements of X3 to be influential on different ft.
Finally, Figure 3.3(f) provides the direction of maximum curvature when directing
influence on a1. Recall that this vector is proportional to the residuals.